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<p><strong>Description:</strong> An introduction to proof techniques, covering proof by contradiction and induction, with an emphasis on the inductive techniques used in proof by induction.</p> <p><strong>Speaker:</strong> Tom Leighton</p>
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<p><span m="560">The</span> <span m="680">following</span> <span m="1120">content</span> <span m="1710">is</span> <span m="1830">provided</span> <span m="2270">under</span> <span m="2550">a</span> <span m="2590">Creative</span> <span m="2990">Commons</span> <span m="3400">license.</span> <span m="4510">Your</span> <span m="4700">support</span> <span m="5200">will</span> <span m="5360">help</span> <span m="5600">MIT</span> <span m="6060">OpenCourseWare</span> <span m="6850">continue</span> <span m="7360">to</span> <span m="7440">offer</span> <span m="7850">high</span> <span m="8090">quality</span> <span m="8610">educational</span> <span m="9240">resources</span> <span m="9860">for</span> <span m="10010">free.</span> <span m="11220">To</span> <span m="11320">make</span> <span m="11430">a</span> <span m="11470">donation,</span> <span m="12160">or</span> <span m="12430">view</span> <span m="12870">additional</span> <span m="13290">materials</span> <span m="13820">from</span> <span m="13980">hundreds</span> <span m="14410">of</span> <span m="14520">MIT</span> <span m="14950">courses,</span> <span m="16059">visit</span> <span m="16280">MIT</span> <span m="16700">OpenCourseWare</span> <span m="17740">at</span> <span m="17910">ocw.mit.edu.</span> </p>
<p><span m="23730">PROFESSOR: Last</span> <span m="24070">week</span> <span m="24230">we</span> <span m="24350">talked</span> <span m="24670">about</span> <span m="24850">the</span> <span m="24950">key</span> <span m="25740">components</span> <span m="26310">of</span> <span m="26400">a proof,</span> <span m="27290">propositions,</span> <span m="28560">axioms,</span> <span m="29470">and</span> <span m="29630">logical</span> <span m="30030">deductions,</span> <span m="31380">and</span> <span m="31520">as</span> <span m="31590">you</span> <span m="31700">probably</span> <span m="32070">talked</span> <span m="32369">about</span> <span m="32619">during</span> <span m="32830">recitation,</span> <span m="34580">we're</span> <span m="34790">not</span> <span m="35000">going</span> <span m="35100">to</span> <span m="35170">worry</span> <span m="35570">too</span> <span m="35830">much</span> <span m="36180">about</span> <span m="36600">what</span> <span m="36830">axioms</span> <span m="37690">or</span> <span m="37860">logical</span> <span m="38370">deductions</span> <span m="38930">that</span> <span m="39010">you</span> <span m="39120">use.</span> <span m="40360">Anything</span> <span m="41360">that</span> <span m="41480">is</span> <span m="41630">reasonable,</span> <span m="42170">is</span> <span m="42290">fine</span> <span m="42560">by</span> <span m="42760">us.</span> <span m="43490">We're</span> <span m="43640">not</span> <span m="43830">going</span> <span m="43940">to</span> <span m="44020">ask</span> <span m="44380">you</span> <span m="44470">to</span> <span m="44570">know</span> <span m="44760">what</span> <span m="44910">modus</span> <span m="45280">ponens</span> <span m="45810">is,</span> <span m="46240">or</span> <span m="46340">to</span> <span m="46420">label</span> <span m="46920">some</span> <span m="47200">law</span> <span m="47580">when</span> <span m="47700">you</span> <span m="47790">make</span> <span m="48000">a</span> <span m="48050">logical</span> <span m="48470">deduction.</span> <span m="49420">Just,</span> <span m="49970">you know,</span> <span m="50280">be</span> <span m="50470">reasonable.</span> <span m="51360">Any</span> <span m="51680">facts</span> <span m="52070">you</span> <span m="52190">knew</span> <span m="52450">coming</span> <span m="52790">into</span> <span m="52930">this</span> <span m="53170">course</span> <span m="53500">about</span> <span m="53690">mathematics,</span> <span m="54860">probably</span> <span m="55170">close</span> <span m="55510">enough</span> <span m="55650">to</span> <span m="55740">use</span> <span m="56200">as</span> <span m="56380">an</span> <span m="56480">axiom.</span> <span m="57470">Want</span> <span m="57590">to</span> <span m="57630">make</span> <span m="57770">sure</span> <span m="57950">your</span> <span m="58080">axioms</span> <span m="58480">are</span> <span m="58550">consistent,</span> <span m="59990">but</span> <span m="60360">that's</span> <span m="60600">OK.</span> </p>
<p><span m="61440">Now</span> <span m="61570">the</span> <span m="61700">exception</span> <span m="62240">to</span> <span m="62290">this</span> <span m="62500">would</span> <span m="62630">be,</span> <span m="63610">is</span> <span m="63760">say</span> <span m="63980">we're</span> <span m="64129">on</span> <span m="64209">an</span> <span m="64280">exam,</span> <span m="65160">and</span> <span m="65459">we</span> <span m="65630">ask</span> <span m="65910">you</span> <span m="66010">to</span> <span m="66100">prove</span> <span m="66500">some</span> <span m="66930">proposition,</span> <span m="68080">p.</span> <span m="69080">Well</span> <span m="69340">you</span> <span m="69490">can't</span> <span m="69820">say,</span> <span m="70220">I</span> <span m="70400">already</span> <span m="70620">knew</span> <span m="70850">p,</span> <span m="71420">it's</span> <span m="71620">an</span> <span m="71700">axiom,</span> <span m="72970">check.</span> <span m="73730">That's</span> <span m="74020">not</span> <span m="74210">so</span> <span m="74330">good.</span> <span m="74550">We're</span> <span m="74700">asking</span> <span m="75100">you</span> <span m="75220">to</span> <span m="75610">prove</span> <span m="75890">it</span> <span m="76010">from</span> <span m="76230">some</span> <span m="76480">more</span> <span m="76710">elementary</span> <span m="77500">facts.</span> <span m="80060">OK</span> <span m="80460">it's</span> <span m="80610">also</span> <span m="80980">don't</span> <span m="81150">want</span> <span m="81290">me</span> <span m="81380">making</span> <span m="81660">wild</span> <span m="82060">leaps</span> <span m="82390">of</span> <span m="82500">faith,</span> <span m="82980">or</span> <span m="83170">saying</span> <span m="83590">it's</span> <span m="83770">obvious</span> <span m="84330">that,</span> <span m="84590">unless</span> <span m="84820">it</span> <span m="84930">really</span> <span m="85470">is</span> <span m="85650">obvious.</span> <span m="86130">That</span> <span m="86300">kind</span> <span m="86460">of</span> <span m="86520">stuff</span> <span m="86770">can get you in</span> <span m="87180">trouble.</span> <span m="87920">Much</span> <span m="88170">better</span> <span m="88370">to</span> <span m="88590">sort</span> <span m="88760">of</span> <span m="88810">explain</span> <span m="89370">the</span> <span m="89480">reasoning</span> <span m="90360">in</span> <span m="90500">the</span> <span m="90590">proof.</span> </p>
<p><span m="92470">Now</span> <span m="92770">the</span> <span m="92900">proofs</span> <span m="93170">that</span> <span m="93290">we</span> <span m="93410">covered</span> <span m="94040">last</span> <span m="94390">week</span> <span m="94650">in</span> <span m="94730">recitation,</span> <span m="95480">the</span> <span m="95570">problems</span> <span m="96060">set,</span> <span m="96610">were</span> <span m="96780">all</span> <span m="97160">examples</span> <span m="97840">of</span> <span m="97940">what</span> <span m="98070">are</span> <span m="98140">called</span> <span m="98530">direct</span> <span m="99130">proofs.</span> <span m="100140">You</span> <span m="100240">start</span> <span m="100500">with</span> <span m="100580">some</span> <span m="100750">axioms,</span> <span m="101840">you</span> <span m="101980">have</span> <span m="102300">some</span> <span m="102450">theorems</span> <span m="102820">you</span> <span m="102910">knew</span> <span m="103050">before</span> <span m="103370">or</span> <span m="103690">you</span> <span m="103800">proved</span> <span m="104080">along</span> <span m="104340">the</span> <span m="104430">way,</span> <span m="105100">and</span> <span m="105250">you</span> <span m="105340">make</span> <span m="105620">logical</span> <span m="106040">deductions</span> <span m="106590">until</span> <span m="106800">you</span> <span m="106960">get</span> <span m="107300">to</span> <span m="107400">where</span> <span m="107550">you</span> <span m="107660">want</span> <span m="107860">to</span> <span m="107910">go,</span> <span m="108580">the</span> <span m="108680">theorem.</span> <span m="109960">We're</span> <span m="110310">going</span> <span m="110420">to</span> <span m="110590">start</span> <span m="110910">today</span> <span m="111930">with</span> <span m="112580">an</span> <span m="112750">indirect</span> <span m="113360">proof.</span> <span m="113980">For</span> <span m="114050">example,</span> <span m="114470">a</span> <span m="114530">proof</span> <span m="114890">by</span> <span m="115020">contradiction.</span> <span m="116360">And</span> <span m="116490">this</span> <span m="116650">is</span> <span m="116740">a</span> <span m="116800">little</span> <span m="117060">bit</span> <span m="117270">different.</span> <span m="118320">In</span> <span m="118420">a</span> <span m="118520">proof</span> <span m="118740">by</span> <span m="118870">contradiction,</span> <span m="120860">you</span> <span m="121070">assume</span> <span m="121580">the</span> <span m="121900">opposite</span> <span m="122740">of</span> <span m="122880">what</span> <span m="123050">you're</span> <span m="123170">trying</span> <span m="123540">to</span> <span m="123620">prove.</span> <span m="125350">Then</span> <span m="125550">you</span> <span m="125640">just</span> <span m="125950">take</span> <span m="126170">steps</span> <span m="126490">for</span> <span m="126750">logical</span> <span m="127120">deductions</span> <span m="127570">forward</span> <span m="127980">until</span> <span m="128340">you</span> <span m="128500">arrive</span> <span m="128960">at</span> <span m="129020">a</span> <span m="129080">contradiction,</span> <span m="130610">something</span> <span m="131160">where</span> <span m="131250">you</span> <span m="131400">prove</span> <span m="131720">false</span> <span m="132090">equals</span> <span m="132390">true.</span> </p>
<p><span m="133940">Now</span> <span m="134320">if</span> <span m="134470">you</span> <span m="134560">can</span> <span m="134670">ever</span> <span m="134850">get</span> <span m="135010">to</span> <span m="135070">the</span> <span m="135170">point</span> <span m="135420">we</span> <span m="135540">approve</span> <span m="136030">something</span> <span m="136350">is</span> <span m="136500">false</span> <span m="137510">and</span> <span m="137810">true,</span> <span m="138830">that</span> <span m="139040">means</span> <span m="139260">what</span> <span m="139390">you</span> <span m="139490">assumed</span> <span m="139810">at</span> <span m="139870">the</span> <span m="139970">start</span> <span m="140220">had</span> <span m="140370">to</span> <span m="140430">be</span> <span m="140540">wrong.</span> <span m="141750">Namely,</span> <span m="142640">what</span> <span m="143000">you're</span> <span m="143100">trying</span> <span m="143310">to</span> <span m="143390">prove</span> <span m="144060">has</span> <span m="144310">to</span> <span m="144400">be</span> <span m="144510">true.</span> <span m="144920">So</span> <span m="145140">let's</span> <span m="145320">write</span> <span m="145520">that</span> <span m="145720">down,</span> <span m="145940">because</span> <span m="146090">it</span> <span m="146180">can</span> <span m="146470">be</span> <span m="147350">a</span> <span m="147430">little</span> <span m="147620">confusing</span> <span m="148080">the</span> <span m="148160">first</span> <span m="148440">time.</span> </p>
<p><span m="160200">So</span> <span m="160640">to</span> <span m="160730">prove</span> <span m="162590">a</span> <span m="162650">proposition</span> <span m="163340">p</span> <span m="163860">is</span> <span m="163980">true,</span> <span m="169000">you</span> <span m="169120">will</span> <span m="169560">assume</span> <span m="173430">that</span> <span m="173600">it's</span> <span m="173740">false.</span> <span m="179430">In</span> <span m="179580">other</span> <span m="179770">words,</span> <span m="180170">that</span> <span m="180340">not</span> <span m="180750">p</span> <span m="180960">is</span> <span m="181070">true.</span> <span m="190210">And</span> <span m="190440">then</span> <span m="193560">you</span> <span m="193760">use</span> <span m="194270">that</span> <span m="194480">hypothesis,</span> <span m="199060">namely</span> <span m="199340">the</span> <span m="199460">p</span> <span m="199630">is</span> <span m="199800">false,</span> <span m="202470">to</span> <span m="202640">derive</span> <span m="205310">a</span> <span m="205420">falsehood.</span> <span m="207040">In other words,</span> <span m="207300">you</span> <span m="207390">prove</span> <span m="207570">a</span> <span m="207630">falsehood</span> <span m="208120">is</span> <span m="208220">true.</span> <span m="209590">And</span> <span m="209740">this</span> <span m="209950">is</span> <span m="210950">called</span> <span m="211300">deriving</span> <span m="211830">a</span> <span m="211900">contradiction.</span> </p>
<p><span m="219350">And</span> <span m="219440">so</span> <span m="219570">it</span> <span m="219670">must</span> <span m="220010">be</span> <span m="220200">that,</span> <span m="220310">in</span> <span m="220430">fact,</span> <span m="220890">p</span> <span m="221000">is</span> <span m="221130">not</span> <span m="221360">false,</span> <span m="221720">namely</span> <span m="222000">that</span> <span m="222120">it's</span> <span m="222320">true.</span> <span m="223980">Now this works</span> <span m="225070">because</span> <span m="225710">if</span> <span m="225840">you</span> <span m="226010">can</span> <span m="226160">prove,</span> <span m="228410">if</span> <span m="228600">not</span> <span m="228970">p</span> <span m="231490">implies</span> <span m="232090">false</span> <span m="233460">is</span> <span m="233620">true,</span> <span m="237290">well</span> <span m="237490">from</span> <span m="237650">last</span> <span m="237970">time,</span> <span m="239660">the</span> <span m="239830">only</span> <span m="240050">way</span> <span m="240290">this</span> <span m="240640">is</span> <span m="240750">a</span> <span m="240810">true</span> <span m="241060">statement</span> <span m="243440">is</span> <span m="243680">if</span> <span m="243820">this</span> <span m="244020">is</span> <span m="244170">false,</span> <span m="245530">namely</span> <span m="246380">p</span> <span m="246570">is</span> <span m="246700">true.</span> <span m="249020">All</span> <span m="249130">right?</span> <span m="249240">So</span> <span m="249350">we</span> <span m="249450">can</span> <span m="249590">conclude</span> <span m="250180">that</span> <span m="250290">p</span> <span m="250520">is</span> <span m="250650">true,</span> <span m="251520">if</span> <span m="251730">we</span> <span m="251830">can</span> <span m="251950">show</span> <span m="252230">that</span> <span m="252420">not</span> <span m="252750">P</span> <span m="252990">implies</span> <span m="254080">a</span> <span m="254150">falsehood.</span> <span m="256850">Any</span> <span m="257010">questions</span> <span m="258440">about</span> <span m="258640">that?</span> <span m="258779">It's</span> <span m="258839">sort</span> <span m="259040">of</span> <span m="259130">a</span> <span m="259630">lot</span> <span m="259890">of</span> <span m="260170">sort</span> <span m="260339">of</span> <span m="260399">notation,</span> <span m="261860">and</span> <span m="262340">until</span> <span m="262460">you've</span> <span m="262610">seen,</span> <span m="262970">it</span> <span m="263060">can</span> <span m="263280">be</span> <span m="263390">confusing.</span> </p>
<p><span m="265300">So maybe</span> <span m="265420">we</span> <span m="265510">should</span> <span m="265720">do</span> <span m="266560">an</span> <span m="266640">example.</span> <span m="268970">Let's</span> <span m="269180">prove</span> <span m="271440">that</span> <span m="271640">square</span> <span m="271990">root</span> <span m="272130">of</span> <span m="272210">2</span> <span m="272490">is</span> <span m="272590">irrational.</span> <span m="280330">Is</span> <span m="280440">irrational.</span> <span m="285540">OK,</span> <span m="285830">everybody</span> <span m="286110">knows</span> <span m="286320">what</span> <span m="286410">an</span> <span m="286480">irrational</span> <span m="287060">number</span> <span m="287410">is?</span> <span m="289290">That's</span> <span m="289470">something</span> <span m="289770">that</span> <span m="290070">can't</span> <span m="290480">be</span> <span m="290590">expressed</span> <span m="291090">as</span> <span m="291200">the</span> <span m="291320">ratio</span> <span m="291740">of</span> <span m="291830">integers.</span> <span m="293460">OK,</span> <span m="293630">and</span> <span m="293700">probably</span> <span m="293970">most</span> <span m="294270">people</span> <span m="294510">already</span> <span m="294820">know</span> <span m="295120">that--</span> <span m="295440">how</span> <span m="295620">many</span> <span m="295780">people</span> <span m="296210">have</span> <span m="296990">not</span> <span m="297300">seen</span> <span m="297460">a proof</span> <span m="297840">that</span> <span m="297940">square</span> <span m="298150">root</span> <span m="298240">of</span> <span m="298330">2</span> <span m="298690">is</span> <span m="298800">a</span> <span m="298870">rational</span> <span m="299300">before?</span> <span m="301780">So</span> <span m="301920">most</span> <span m="302220">of</span> <span m="302310">you</span> <span m="302450">have</span> <span m="302550">seen</span> <span m="302730">a proof of</span> <span m="303050">that,</span> <span m="303300">good.</span> <span m="303820">You know, if</span> <span m="304070">you</span> <span m="304150">try</span> <span m="304340">to</span> <span m="304420">do</span> <span m="304550">a</span> <span m="304610">direct</span> <span m="305060">proof</span> <span m="305240">for</span> <span m="305320">this,</span> <span m="305520">it's</span> <span m="305640">pretty</span> <span m="305850">hard.</span> <span m="306840">How</span> <span m="307030">do</span> <span m="307100">you</span> <span m="307190">show</span> <span m="307530">there's</span> <span m="307760">no</span> <span m="308060">way</span> <span m="308350">to</span> <span m="308460">represent</span> <span m="308900">the</span> <span m="308980">square</span> <span m="309160">root</span> <span m="309280">of</span> <span m="309390">2</span> <span m="309690">is</span> <span m="310560">as</span> <span m="310740">integers</span> <span m="311120">a</span> <span m="311260">over</span> <span m="311450">b?</span> </p>
<p><span m="312950">But</span> <span m="313070">it's</span> <span m="313210">very</span> <span m="313510">easy,</span> <span m="314720">if</span> <span m="314900">we</span> <span m="314990">do</span> <span m="315150">a</span> <span m="315220">proof</span> <span m="315470">by</span> <span m="315590">contradiction.</span> <span m="317040">Now</span> <span m="317160">when</span> <span m="317240">you're</span> <span m="317310">doing</span> <span m="317560">a</span> <span m="317620">proof</span> <span m="317840">by</span> <span m="317960">contradiction,</span> <span m="319010">always</span> <span m="319390">start</span> <span m="319670">off</span> <span m="319890">by</span> <span m="320348">saying,</span> <span m="322180">by</span> <span m="322640">contradiction.</span> <span m="323540">Write</span> <span m="323790">that</span> <span m="324000">down.</span> <span m="325910">And</span> <span m="326120">then</span> <span m="326280">what</span> <span m="326400">you</span> <span m="326700">do</span> <span m="326840">next</span> <span m="327380">is</span> <span m="327720">you</span> <span m="327830">say,</span> <span m="328140">I'm</span> <span m="328290">going</span> <span m="328480">to</span> <span m="328550">assume,</span> <span m="330780">for</span> <span m="330940">the</span> <span m="331060">purpose</span> <span m="331430">of</span> <span m="331520">contradiction,</span> <span m="341690">that</span> <span m="345250">p</span> <span m="345490">is</span> <span m="345660">false</span> <span m="345980">and</span> <span m="346080">when</span> <span m="346160">not</span> <span m="346430">p</span> <span m="346550">is</span> <span m="346690">true.</span> <span m="347350">In</span> <span m="347450">this</span> <span m="347530">case,</span> <span m="347870">that</span> <span m="347950">would</span> <span m="348070">be</span> <span m="349050">square</span> <span m="349260">root</span> <span m="349330">of</span> <span m="349420">2</span> <span m="349900">is</span> <span m="350090">rational.</span> </p>
<p><span m="356170">So</span> <span m="356340">in</span> <span m="356420">this</span> <span m="356660">case,</span> <span m="358350">here's</span> <span m="358600">what</span> <span m="358670">we're</span> <span m="358800">trying</span> <span m="359060">to</span> <span m="359130">prove.</span> <span m="359520">That's</span> <span m="359790">p.</span> <span m="361730">I'm</span> <span m="361890">going</span> <span m="362140">to</span> <span m="362250">assume</span> <span m="363060">not</span> <span m="363380">p,</span> <span m="363590">namely</span> <span m="363930">that</span> <span m="364040">square</span> <span m="364320">root</span> <span m="364410">of</span> <span m="364490">2</span> <span m="364740">is</span> <span m="364880">irrational</span> <span m="365470">number.</span> <span m="366570">Then</span> <span m="366750">I'm</span> <span m="366830">going</span> <span m="366930">to</span> <span m="366990">get</span> <span m="367160">a</span> <span m="367220">contradiction</span> <span m="367870">or</span> <span m="367990">falsehood,</span> <span m="368830">and</span> <span m="368990">then</span> <span m="369110">I'm</span> <span m="369200">going</span> <span m="369320">to</span> <span m="369420">know</span> <span m="371020">that</span> <span m="371150">p</span> <span m="371390">was</span> <span m="371540">true</span> <span m="371790">after</span> <span m="372080">all.</span> <span m="374540">All right, so</span> <span m="374890">let's</span> <span m="375050">see</span> <span m="375190">where</span> <span m="375290">this</span> <span m="375510">leads</span> <span m="375800">us.</span> <span m="378250">Well,</span> <span m="378600">if</span> <span m="378750">square</span> <span m="379000">root</span> <span m="379070">of</span> <span m="379160">2</span> <span m="379330">is</span> <span m="379440">rational</span> <span m="379980">then</span> <span m="380080">we</span> <span m="380180">can</span> <span m="380310">express</span> <span m="380800">it</span> <span m="380970">as</span> <span m="381330">a</span> <span m="381530">over</span> <span m="381740">b,</span> <span m="383070">where</span> <span m="383390">a</span> <span m="383570">over</span> <span m="383720">b</span> <span m="383850">is</span> <span m="383990">a</span> <span m="384050">fraction</span> <span m="384590">in</span> <span m="384690">lowest</span> <span m="385040">terms.</span> <span m="390520">That</span> <span m="390700">means</span> <span m="390990">a and</span> <span m="391240">b</span> <span m="391400">have</span> <span m="391570">no</span> <span m="391760">common</span> <span m="392060">divisors.</span> <span m="396140">And</span> <span m="396580">then</span> <span m="396740">I</span> <span m="396830">can</span> <span m="397020">square</span> <span m="397480">both</span> <span m="397710">sides,</span> <span m="399720">and I</span> <span m="399880">get</span> <span m="400090">two</span> <span m="400360">is</span> <span m="400480">a</span> <span m="400660">squared</span> <span m="401190">over</span> <span m="401400">b</span> <span m="401590">squared.</span> <span m="405370">Then</span> <span m="405470">I</span> <span m="405540">multiply</span> <span m="405980">by</span> <span m="406160">b</span> <span m="406380">squared,</span> <span m="406695">and</span> <span m="407010">I</span> <span m="407070">get</span> <span m="408100">2b</span> <span m="408570">squared</span> <span m="409080">equals</span> <span m="409470">a</span> <span m="409600">squared.</span> </p>
<p><span m="411900">And</span> <span m="411970">what</span> <span m="412120">does</span> <span m="412270">that</span> <span m="412650">imply</span> <span m="413010">about</span> <span m="413270">a?</span> <span m="415340">What</span> <span m="415480">can</span> <span m="415600">you</span> <span m="415690">tell</span> <span m="415900">me</span> <span m="416020">about</span> <span m="416360">a if it</span> <span m="416840">equals--</span> <span m="417710">if</span> <span m="417970">a</span> <span m="418070">squared</span> <span m="418340">is</span> <span m="418430">2b</span> <span m="418790">squared?</span> <span m="424840">Anything</span> <span m="425160">special</span> <span m="425540">about</span> <span m="425780">a?</span> <span m="425910">Could</span> <span m="426120">a</span> <span m="426330">be</span> <span m="426990">anything?</span> <span m="429990">A</span> <span m="430180">squared</span> <span m="430510">is</span> <span m="430670">even,</span> <span m="432040">all right?</span> <span m="432180">Because 2b</span> <span m="432480">squared</span> <span m="432760">is</span> <span m="432840">an</span> <span m="432920">even</span> <span m="433110">number,</span> <span m="433380">so</span> <span m="433570">a</span> <span m="433770">squared</span> <span m="434080">is</span> <span m="434190">even,</span> <span m="435360">and</span> <span m="435490">what</span> <span m="435600">does</span> <span m="435680">that</span> <span m="435810">mean</span> <span m="435960">about</span> <span m="436180">a?</span> <span m="438605">A</span> <span m="439090">is</span> <span m="439270">even.</span> <span m="440140">B</span> <span m="440320">squared</span> <span m="440720">is</span> <span m="441045">even,</span> <span m="441126">then</span> <span m="441207">a</span> <span m="441288">is</span> <span m="441370">even.</span> <span m="443570">So</span> <span m="443900">a</span> <span m="443970">is</span> <span m="444130">even.</span> <span m="446860">So</span> <span m="447260">we</span> <span m="447420">could</span> <span m="447540">write</span> <span m="447810">that</span> <span m="448240">as</span> <span m="448690">two</span> <span m="449110">divides</span> <span m="449730">a.</span> <span m="450080">That's</span> <span m="450340">the</span> <span m="450440">divide</span> <span m="450890">symbol.</span> <span m="451250">You'll</span> <span m="451360">see</span> <span m="451530">a</span> <span m="451600">lot</span> <span m="451860">of</span> <span m="451930">that</span> <span m="452300">next</span> <span m="452560">week.</span> </p>
<p><span m="454700">All</span> <span m="454800">right</span> <span m="454920">if</span> <span m="455060">a</span> <span m="455310">is</span> <span m="455440">even,</span> <span m="455800">what</span> <span m="455930">do</span> <span m="456010">I</span> <span m="456080">know</span> <span m="456270">about</span> <span m="456470">a</span> <span m="456600">squared?</span> <span m="457940">I</span> <span m="458030">know</span> <span m="458200">more</span> <span m="458450">than</span> <span m="458580">just,</span> <span m="458840">it's</span> <span m="458960">even.</span> <span m="464020">What</span> <span m="464200">is it?</span> <span m="464510">It's</span> <span m="464550">multiple</span> <span m="464990">of</span> <span m="465810">4,</span> <span m="466490">yeah.</span> <span m="467160">So</span> <span m="467330">that</span> <span m="467550">means</span> <span m="467870">that</span> <span m="469110">four</span> <span m="471070">divides</span> <span m="471550">a</span> <span m="471710">squared.</span> <span m="473030">A</span> <span m="473190">squared</span> <span m="473500">is</span> <span m="473790">2b</span> <span m="474230">squared,</span> <span m="474740">so</span> <span m="474840">that</span> <span m="474870">means</span> <span m="475150">that</span> <span m="475350">four</span> <span m="475830">divides</span> <span m="476370">2b</span> <span m="476840">squared.</span> <span m="477970">Divide</span> <span m="478300">each</span> <span m="478460">side</span> <span m="478710">by</span> <span m="478850">2</span> <span m="479250">means</span> <span m="479720">2</span> <span m="480140">divides</span> <span m="480730">b</span> <span m="480970">squared.</span> <span m="482780">What</span> <span m="483000">does</span> <span m="483100">that</span> <span m="483230">imply</span> <span m="483520">about</span> <span m="483940">b?</span> <span m="487764">B is</span> <span m="488250">even.</span> <span m="490870">All right,</span> <span m="491110">b is</span> <span m="491280">even,</span> <span m="492200">good.</span> <span m="493750">Well,</span> <span m="494830">I've</span> <span m="494960">got</span> <span m="495240">a</span> <span m="495500">is</span> <span m="495670">even</span> <span m="496030">and</span> <span m="496300">b</span> <span m="496600">is</span> <span m="496760">even.</span> <span m="499620">I</span> <span m="499730">got</span> <span m="499930">a</span> <span m="499980">contradiction</span> <span m="500610">here,</span> <span m="500930">somewhere?</span> <span m="504080">Yeah</span> <span m="504910">a</span> <span m="505390">over</span> <span m="505670">b</span> <span m="505920">was</span> <span m="506160">not</span> <span m="506700">a</span> <span m="506810">fraction</span> <span m="507320">in</span> <span m="507420">lowest</span> <span m="507790">terms,</span> <span m="508260">because</span> <span m="508710">both</span> <span m="509130">and</span> <span m="509500">b</span> <span m="509730">are</span> <span m="509840">even.</span> <span m="511410">All</span> <span m="511470">right,</span> <span m="511850">so</span> <span m="513080">that</span> <span m="513270">implies</span> <span m="516690">a</span> <span m="517030">over</span> <span m="517299">b</span> <span m="518600">is</span> <span m="518909">not</span> <span m="520760">in</span> <span m="520919">lowest</span> <span m="521280">terms.</span> <span m="526060">And</span> <span m="526400">that</span> <span m="528120">is</span> <span m="528390">a</span> <span m="528470">contradiction.</span> </p>
<p><span m="530500">Now</span> <span m="530840">you'll</span> <span m="530990">see</span> <span m="531180">that</span> <span m="531400">written</span> <span m="531590">lots</span> <span m="531980">of</span> <span m="532060">ways.</span> <span m="532660">One</span> <span m="532780">is,</span> <span m="532900">you</span> <span m="533360">can</span> <span m="533520">say a</span> <span m="533780">contradiction,</span> <span m="534410">sometimes</span> <span m="534820">you'll</span> <span m="534960">see</span> <span m="535310">just</span> <span m="535635">this</span> <span m="536280">sharp</span> <span m="536670">symbol</span> <span m="537040">written,</span> <span m="537880">and</span> <span m="537990">that</span> <span m="538150">means</span> <span m="538360">you've</span> <span m="538520">got</span> <span m="538740">to</span> <span m="538820">a</span> <span m="538880">contradiction.</span> <span m="539460">Because</span> <span m="540480">here,</span> <span m="540810">we</span> <span m="541490">had</span> <span m="541730">it</span> <span m="541820">being</span> <span m="542160">in</span> <span m="542240">lowest</span> <span m="542590">terms,</span> <span m="543510">and</span> <span m="543630">here</span> <span m="543870">we</span> <span m="543990">have</span> <span m="544250">it</span> <span m="544380">not</span> <span m="544590">in</span> <span m="544680">lowest</span> <span m="544980">terms.</span> <span m="545260">You</span> <span m="545350">can't</span> <span m="545590">have</span> <span m="545780">both</span> <span m="546020">at</span> <span m="546100">the</span> <span m="546170">same</span> <span m="546440">time,</span> <span m="546700">so</span> <span m="546820">you</span> <span m="546860">got</span> <span m="546990">a</span> <span m="547040">contradiction.</span> <span m="548910">And</span> <span m="549160">that</span> <span m="549400">means</span> <span m="549700">we've</span> <span m="549890">now</span> <span m="550510">proved</span> <span m="551690">that</span> <span m="551850">this</span> <span m="552010">assumption</span> <span m="552500">was</span> <span m="552650">wrong.</span> <span m="554080">Square</span> <span m="554310">root</span> <span m="554430">of</span> <span m="554520">2</span> <span m="554730">is</span> <span m="556050">not</span> <span m="556240">rational,</span> <span m="556750">so</span> <span m="556850">it</span> <span m="556950">must</span> <span m="557000">be</span> <span m="557490">irrational.</span> <span m="564420">and</span> <span m="564550">then</span> <span m="564720">we</span> <span m="564830">put</span> <span m="565000">a</span> <span m="565050">little</span> <span m="565250">box</span> <span m="565710">here</span> <span m="565890">at</span> <span m="565970">the</span> <span m="566130">end,</span> <span m="566540">or</span> <span m="566620">sometimes</span> <span m="566990">you'll</span> <span m="567110">see</span> <span m="567270">a</span> <span m="567340">check,</span> <span m="568120">sometimes</span> <span m="568510">you'll</span> <span m="568620">see</span> <span m="568810">QED,</span> <span m="570000">that</span> <span m="570160">says</span> <span m="570650">the</span> <span m="570740">proof's</span> <span m="571020">done</span> <span m="571230">now.</span> <span m="572910">It's</span> <span m="573070">over.</span> <span m="574640">Any</span> <span m="574810">questions</span> <span m="577000">about</span> <span m="577220">that</span> <span m="577410">proof?</span> </p>
<p><span m="578790">We're</span> <span m="578890">going</span> <span m="578970">to</span> <span m="579010">do</span> <span m="579110">a</span> <span m="579220">lot</span> <span m="579490">of</span> <span m="579570">proofs</span> <span m="579800">by</span> <span m="579920">contradiction.</span> <span m="581740">Actually,</span> <span m="582530">there's</span> <span m="582730">an</span> <span m="582810">interesting</span> <span m="583190">story</span> <span m="583640">behind</span> <span m="584040">this</span> <span m="584220">proof.</span> <span m="585630">As</span> <span m="585830">far</span> <span m="586060">as</span> <span m="586150">we</span> <span m="586260">know,</span> <span m="586500">it</span> <span m="586590">was</span> <span m="586750">first</span> <span m="587080">discovered</span> <span m="587520">by</span> <span m="587650">the</span> <span m="587930">Pythagoreans,</span> <span m="589600">way</span> <span m="589930">back</span> <span m="590200">when</span> <span m="590470">in</span> <span m="590630">ancient</span> <span m="590990">Greece.</span> <span m="592270">And</span> <span m="592850">the</span> <span m="592970">Pythagoreans</span> <span m="593770">were</span> <span m="594070">a</span> <span m="594190">religious</span> <span m="594620">society</span> <span m="595330">started</span> <span m="595650">by</span> <span m="595820">Pythagoras,</span> <span m="596510">of</span> <span m="596980">Pythagoras</span> <span m="597550">theorem</span> <span m="597940">frame.</span> <span m="599350">Now</span> <span m="599600">it</span> <span m="599710">sounds</span> <span m="600060">weird</span> <span m="600430">today,</span> <span m="601260">but</span> <span m="601470">back</span> <span m="601750">then,</span> <span m="602840">in</span> <span m="602970">ancient</span> <span m="603240">Greece,</span> <span m="603850">math</span> <span m="604250">was</span> <span m="604430">a</span> <span m="604530">religion,</span> <span m="606020">all</span> <span m="606180">right?</span> <span m="607080">Every</span> <span m="607250">once</span> <span m="607480">in</span> <span m="607530">while</span> <span m="607650">you'll</span> <span m="607770">see</span> <span m="607900">somebody</span> <span m="608300">around</span> <span m="608490">MIT,</span> <span m="608970">and</span> <span m="609210">you'll</span> <span m="609350">think</span> <span m="609670">he</span> <span m="609820">must</span> <span m="610070">think</span> <span m="610230">math</span> <span m="610460">is</span> <span m="610580">a</span> <span m="610640">religion,</span> <span m="611920">but</span> <span m="612230">back</span> <span m="612490">then</span> <span m="612640">it</span> <span m="612720">really</span> <span m="612990">was,</span> <span m="613510">and</span> <span m="613670">it</span> <span m="613740">was</span> <span m="613870">ruled</span> <span m="614180">by</span> <span m="614300">God,</span> <span m="614860">because</span> <span m="615080">this</span> <span m="615250">is</span> <span m="615460">ancient</span> <span m="615800">Greece.</span> </p>
<p><span m="616620">And</span> <span m="616840">there</span> <span m="616940">were</span> <span m="617060">two</span> <span m="617280">key</span> <span m="617500">gods</span> <span m="618230">in</span> <span m="618340">this</span> <span m="618510">religion,</span> <span m="619770">Apeiron</span> <span m="620430">and</span> <span m="620610">Peros.</span> <span m="621790">Now</span> <span m="622040">Apeiron</span> <span m="622990">was</span> <span m="623280">the</span> <span m="623360">bad</span> <span m="623730">god,</span> <span m="624750">and</span> <span m="625140">he</span> <span m="625320">was</span> <span m="625480">the</span> <span m="625600">god</span> <span m="625910">of</span> <span m="626000">infinity,</span> <span m="626730">because</span> <span m="627020">infinity</span> <span m="627500">was</span> <span m="627660">considered</span> <span m="628070">all</span> <span m="628370">that</span> <span m="628480">was</span> <span m="628640">bad.</span> <span m="629700">And</span> <span m="630030">I</span> <span m="630180">don't</span> <span m="630320">think</span> <span m="630470">we'll</span> <span m="630590">do</span> <span m="630720">a</span> <span m="630790">lot</span> <span m="630940">with</span> <span m="631060">infinity</span> <span m="631540">this</span> <span m="631640">term,</span> <span m="632020">but</span> <span m="632120">if</span> <span m="632220">we</span> <span m="632230">do</span> <span m="632510">you'll</span> <span m="632890">appreciate</span> <span m="633570">why</span> <span m="633860">that's</span> <span m="634090">the</span> <span m="634170">case.</span> <span m="635270">And</span> <span m="636140">Peros</span> <span m="636700">was</span> <span m="636900">a</span> <span m="636970">good</span> <span m="637170">god.</span> <span m="637540">He</span> <span m="637780">was</span> <span m="637930">the</span> <span m="638040">god</span> <span m="638330">of</span> <span m="638390">the</span> <span m="638470">finite</span> <span m="638940">world,</span> <span m="639580">and</span> <span m="639780">represented</span> <span m="640350">everything</span> <span m="640750">that</span> <span m="640850">was</span> <span m="641020">good</span> <span m="641750">to</span> <span m="641900">the</span> <span m="642010">ancient</span> <span m="642290">Greeks.</span> </p>
<p><span m="643570">Now</span> <span m="643750">one</span> <span m="643930">of</span> <span m="644000">their</span> <span m="644140">main</span> <span m="644660">axioms,</span> <span m="645420">or</span> <span m="645570">beliefs,</span> <span m="646930">was</span> <span m="647310">that</span> <span m="647420">there</span> <span m="647530">were</span> <span m="647690">no</span> <span m="648340">irrational</span> <span m="649110">numbers.</span> <span m="650290">They</span> <span m="650420">just</span> <span m="650920">didn't</span> <span m="651420">exist.</span> <span m="653070">Now</span> <span m="653380">the</span> <span m="653510">reason</span> <span m="653930">is,</span> <span m="654030">they</span> <span m="654130">didn't</span> <span m="654480">like</span> <span m="654730">irrational</span> <span m="655210">numbers.</span> <span m="655570">They</span> <span m="655650">were</span> <span m="655790">bad.</span> <span m="656690">Because,</span> <span m="657660">well</span> <span m="657830">they're</span> <span m="657970">infinite.</span> <span m="658420">You</span> <span m="658530">can't</span> <span m="658910">write</span> <span m="659200">them</span> <span m="660060">as</span> <span m="660240">a</span> <span m="660330">decimal</span> <span m="661390">without</span> <span m="661690">repeating</span> <span m="662140">forever,</span> <span m="663190">and</span> <span m="663330">you know,</span> <span m="663640">just</span> <span m="664140">an</span> <span m="664300">infinite</span> <span m="664990">sort</span> <span m="665130">of</span> <span m="665220">decimal</span> <span m="665540">representation.</span> <span m="667160">They</span> <span m="667320">liked</span> <span m="668590">rational</span> <span m="669100">numbers,</span> <span m="669570">you know, like</span> <span m="669760">one</span> <span m="670000">seventh?</span> <span m="671530">That's</span> <span m="671770">a</span> <span m="671840">good</span> <span m="672060">number,</span> <span m="672420">because</span> <span m="673010">you</span> <span m="673130">can</span> <span m="673250">write</span> <span m="673490">it</span> <span m="673540">as</span> <span m="673650">0.142857</span> <span m="677290">repeating.</span> <span m="679330">So</span> <span m="679420">rational</span> <span m="679910">numbers</span> <span m="680270">are</span> <span m="680450">finite,</span> <span m="681210">in</span> <span m="681290">that</span> <span m="681370">you</span> <span m="681470">can</span> <span m="681580">always</span> <span m="681870">find</span> <span m="682100">a</span> <span m="682140">repeating</span> <span m="682520">pattern</span> <span m="682950">of</span> <span m="683040">finite</span> <span m="683390">length.</span> <span m="684100">Irrational</span> <span m="684700">numbers</span> <span m="685140">are</span> <span m="685270">not.</span> <span m="685540">They're</span> <span m="685700">infinite</span> <span m="686090">in</span> <span m="686180">that</span> <span m="686390">sense</span> <span m="686820">of</span> <span m="686930">the</span> <span m="687360">ancient</span> <span m="687620">Greeks.</span> </p>
<p><span m="689080">So</span> <span m="689240">they</span> <span m="689360">said</span> <span m="689550">there</span> <span m="689650">were</span> <span m="689890">no</span> <span m="690110">irrational</span> <span m="690530">numbers.</span> <span m="690840">That</span> <span m="690990">was</span> <span m="691360">an</span> <span m="691460">early</span> <span m="691710">axiom.</span> <span m="693380">Now</span> <span m="694690">they</span> <span m="694850">also</span> <span m="695090">had</span> <span m="695280">an</span> <span m="695370">axiom</span> <span m="696320">that</span> <span m="696480">said</span> <span m="696810">that</span> <span m="697040">every</span> <span m="698440">length</span> <span m="699170">of</span> <span m="699290">a</span> <span m="699360">line</span> <span m="701030">was</span> <span m="701340">finite,</span> <span m="702530">therefore</span> <span m="702930">rational.</span> <span m="704210">All right,</span> <span m="704380">you know</span> <span m="704520">the</span> <span m="704770">ancient</span> <span m="705080">Greeks</span> <span m="705300">were</span> <span m="705410">good</span> <span m="705650">with</span> <span m="705890">a</span> <span m="706960">compass,</span> <span m="707340">and</span> <span m="707510">drawing</span> <span m="707800">lines</span> <span m="708130">with</span> <span m="708230">straight</span> <span m="708480">edges</span> <span m="708740">and</span> <span m="708850">stuff.</span> <span m="709570">So</span> <span m="709690">they</span> <span m="709820">said</span> <span m="710020">any</span> <span m="710230">line</span> <span m="710510">that you</span> <span m="710730">can</span> <span m="710910">construct</span> <span m="712090">has</span> <span m="712470">a</span> <span m="712670">finite</span> <span m="713190">length</span> <span m="713530">to</span> <span m="713710">it,</span> <span m="713800">so</span> <span m="713920">therefore it</span> <span m="714270">has</span> <span m="714470">a</span> <span m="714540">rational</span> <span m="715130">length.</span> </p>
<p><span m="715390">That</span> <span m="715540">was</span> <span m="715790">axiom</span> <span m="716180">number</span> <span m="716450">two.</span> <span m="718740">Now</span> <span m="719420">they</span> <span m="719540">were</span> <span m="719660">good</span> <span m="719870">geometers,</span> <span m="720410">so</span> <span m="720570">they</span> <span m="720730">knew,</span> <span m="721420">of</span> <span m="721580">course,</span> <span m="721910">they</span> <span m="722010">developed</span> <span m="722380">the</span> <span m="722480">Pythagorean</span> <span m="723050">theorem.</span> <span m="724070">But</span> <span m="724190">if</span> <span m="724270">you</span> <span m="724360">took</span> <span m="724510">a</span> <span m="724580">triangle,</span> <span m="724890">and you</span> <span m="725200">have</span> <span m="725470">side</span> <span m="725760">lengths</span> <span m="726070">one,</span> <span m="727520">the</span> <span m="727630">length</span> <span m="727900">of</span> <span m="727980">the</span> <span m="728050">hypotenuse</span> <span m="729520">was</span> <span m="729590">square</span> <span m="729870">root</span> <span m="730010">2.</span> <span m="731410">Therefore</span> <span m="731750">they</span> <span m="731890">had</span> <span m="732010">a</span> <span m="732070">theorem</span> <span m="732450">that</span> <span m="732640">said</span> <span m="732760">square</span> <span m="733020">root</span> <span m="733180">2</span> <span m="733390">was</span> <span m="733560">rational.</span> <span m="734946">Because</span> <span m="735350">you've</span> <span m="735480">got the</span> <span m="735560">2</span> <span m="735990">axioms there,</span> <span m="736340">right?</span> <span m="738430">Now</span> <span m="739830">eventually,</span> <span m="741170">they</span> <span m="741310">discovered</span> <span m="742630">a</span> <span m="742710">proof</span> <span m="743020">like</span> <span m="743190">that,</span> <span m="743480">that</span> <span m="743580">it</span> <span m="743650">wasn't</span> <span m="744160">rational.</span> <span m="744740">It</span> <span m="744800">was</span> <span m="744930">irrational.</span> </p>
<p><span m="746380">This</span> <span m="746640">caused</span> <span m="746970">quite</span> <span m="747260">a</span> <span m="747320">stir.</span> <span m="748010">First</span> <span m="748420">it</span> <span m="748530">meant</span> <span m="748840">that</span> <span m="749000">their</span> <span m="749120">axioms</span> <span m="749740">were</span> <span m="749920">inconsistent.</span> <span m="751180">Every</span> <span m="751500">theorem</span> <span m="751820">they</span> <span m="751980">proved</span> <span m="752590">was</span> <span m="752780">now</span> <span m="752950">suspect,</span> <span m="754590">once</span> <span m="754750">you</span> <span m="754860">have</span> <span m="755060">inconsistent</span> <span m="755630">axioms.</span> <span m="756600">Even</span> <span m="756930">worse,</span> <span m="758160">the</span> <span m="758310">devil</span> <span m="759940">is</span> <span m="760340">in</span> <span m="760480">their</span> <span m="760610">midst.</span> <span m="761800">Square</span> <span m="762240">root</span> <span m="762410">2</span> <span m="762590">is</span> <span m="762740">infinite</span> <span m="763070">is</span> <span m="763220">the</span> <span m="763320">bad</span> <span m="763860">god,</span> <span m="764300">and</span> <span m="764600">this</span> <span m="764760">is</span> <span m="764860">the</span> <span m="764930">most</span> <span m="765150">basic</span> <span m="765980">length</span> <span m="766320">they</span> <span m="766440">had,</span> <span m="766820">besides</span> <span m="767090">one.</span> <span m="768690">So</span> <span m="768840">that's</span> <span m="769000">a</span> <span m="769040">very</span> <span m="769260">bad</span> <span m="769560">thing.</span> <span m="770300">Sort of</span> <span m="770430">like, you know,</span> <span m="770640">today</span> <span m="771210">we</span> <span m="771300">were</span> <span m="771390">to</span> <span m="771440">wake</span> <span m="771690">up</span> <span m="771820">and</span> <span m="771910">discover</span> <span m="772350">that</span> <span m="773320">there were</span> <span m="773500">only</span> <span m="773940">nine</span> <span m="774310">commandments,</span> <span m="775450">and</span> <span m="775680">the</span> <span m="775790">10th</span> <span m="776240">was</span> <span m="776450">planted</span> <span m="776940">there</span> <span m="777090">by</span> <span m="777250">the</span> <span m="777370">devil.</span> <span m="777950">And</span> <span m="778080">you're</span> <span m="778150">not</span> <span m="778330">sure</span> <span m="778520">which</span> <span m="778710">one,</span> <span m="778930">maybe.</span> <span m="779710">That</span> <span m="780190">would</span> <span m="780310">be</span> <span m="780610">a</span> <span m="780670">big</span> <span m="780970">mess,</span> <span m="781400">sacrilege.</span> </p>
<p><span m="783280">Well,</span> <span m="783460">so</span> <span m="783580">what</span> <span m="783770">were</span> <span m="783860">they</span> <span m="784010">to</span> <span m="784100">do?</span> <span m="784520">This</span> <span m="784710">is</span> <span m="784950">a</span> <span m="785060">disaster</span> <span m="785690">of</span> <span m="786530">major</span> <span m="786850">proportions.</span> <span m="787370">So</span> <span m="788060">they</span> <span m="788180">covered</span> <span m="788510">it</span> <span m="788620">up,</span> <span m="789670">and</span> <span m="789870">they</span> <span m="789940">denied</span> <span m="790300">the</span> <span m="790420">result.</span> <span m="791490">So</span> <span m="791910">they</span> <span m="792270">didn't</span> <span m="792550">want</span> <span m="792680">to</span> <span m="792730">publish</span> <span m="793050">that</span> <span m="793230">proof.</span> <span m="793570">So they</span> <span m="793620">kept</span> <span m="793880">on</span> <span m="794010">saying</span> <span m="794340">square</span> <span m="794590">root</span> <span m="794750">2</span> <span m="794900">was</span> <span m="795050">rational.</span> <span m="796760">But</span> <span m="796910">then,</span> <span m="797530">according</span> <span m="797900">to</span> <span m="798010">legend</span> <span m="798410">there</span> <span m="798500">was</span> <span m="798620">a</span> <span m="798700">Deep</span> <span m="798960">Throat.</span> <span m="800130">Somebody</span> <span m="800760">who</span> <span m="800960">let</span> <span m="801130">out</span> <span m="801330">the</span> <span m="801420">word</span> <span m="802230">and</span> <span m="802410">the</span> <span m="802480">proof</span> <span m="802750">the</span> <span m="802860">square</span> <span m="803120">root</span> <span m="803270">2</span> <span m="803450">is</span> <span m="803550">irrational,</span> <span m="804060">because</span> <span m="804180">that</span> <span m="804320">would</span> <span m="804400">be</span> <span m="804500">very</span> <span m="804770">destabilizing</span> <span m="805630">for</span> <span m="805730">the</span> <span m="805850">society,</span> <span m="807450">and</span> <span m="807590">so</span> <span m="807690">they</span> <span m="807800">killed</span> <span m="808030">them.</span> </p>
<p><span m="809810">This</span> <span m="810000">is</span> <span m="810110">the</span> <span m="810220">legend.</span> <span m="811320">Now,</span> <span m="811400">hard</span> <span m="811600">to</span> <span m="811680">imagine</span> <span m="812030">getting</span> <span m="812200">killed</span> <span m="812500">over</span> <span m="812640">the</span> <span m="812770">irrationality</span> <span m="813430">of</span> <span m="813510">2.</span> <span m="815990">All</span> <span m="816130">right,</span> <span m="816280">we're</span> <span m="816380">going</span> <span m="816460">to</span> <span m="816500">do</span> <span m="816600">a</span> <span m="816660">lot</span> <span m="816860">more</span> <span m="817130">of</span> <span m="817460">these</span> <span m="817670">kind</span> <span m="817840">of</span> <span m="817910">proofs</span> <span m="818260">in</span> <span m="818610">homework</span> <span m="819080">and</span> <span m="819430">throughout</span> <span m="819730">the</span> <span m="819830">term.</span> <span m="820800">The</span> <span m="820890">next</span> <span m="821200">proof</span> <span m="821330">I</span> <span m="821390">want</span> <span m="821580">to</span> <span m="821630">show</span> <span m="821900">you</span> <span m="822750">is</span> <span m="823760">one</span> <span m="823920">of</span> <span m="823970">my</span> <span m="824080">favorite</span> <span m="824470">proof</span> <span m="824680">techniques.</span> <span m="825410">One of</span> <span m="825540">my</span> <span m="825650">favorite</span> <span m="825960">proofs.</span> <span m="826950">And</span> <span m="827240">that</span> <span m="827430">is</span> <span m="827720">a</span> <span m="827880">false</span> <span m="828260">proof.</span> <span m="829420">And</span> <span m="829570">we're</span> <span m="829650">going</span> <span m="829730">to</span> <span m="829970">see</span> <span m="830150">a</span> <span m="830200">lot</span> <span m="830390">of</span> <span m="830440">these</span> <span m="830640">during</span> <span m="830830">the</span> <span m="830920">term,</span> <span m="831190">too.</span> <span m="831370">And if</span> <span m="831510">we</span> <span m="831600">could</span> <span m="831700">bring</span> <span m="831880">the</span> <span m="831970">screens</span> <span m="832340">down.</span> <span m="834030">Somebody</span> <span m="834270">back</span> <span m="834520">there</span> <span m="834660">to--</span> <span m="835260">yeah.</span> <span m="835420">Great.</span> <span m="835730">Bring</span> <span m="835870">the</span> <span m="835940">screens</span> <span m="836280">down,</span> <span m="836530">I'm</span> <span m="836670">going to--</span> <span m="836930">and</span> <span m="837050">then</span> <span m="837180">turn</span> <span m="837370">this</span> <span m="837560">on</span> <span m="838250">for</span> <span m="838600">me.</span> <span m="838750">So</span> <span m="839510">I'm</span> <span m="839730">going</span> <span m="839840">to</span> <span m="839910">prove</span> <span m="840390">to</span> <span m="840490">you</span> <span m="841720">that</span> <span m="841920">90</span> <span m="842520">is</span> <span m="842680">bigger</span> <span m="843000">than</span> <span m="843170">92.</span> <span m="845370">All right?</span> <span m="845660">And</span> <span m="845780">that</span> <span m="846980">hopefully,</span> <span m="847230">is</span> <span m="847310">not</span> <span m="847530">really</span> <span m="847820">true.</span> <span m="849360">But,</span> <span m="849790">you know,</span> <span m="849860">watch</span> <span m="850100">this</span> <span m="850280">proof</span> <span m="851210">and</span> <span m="851360">see</span> <span m="851600">if</span> <span m="853280">there's</span> <span m="853470">any</span> <span m="853640">problems</span> <span m="854100">with</span> <span m="854250">it.</span> </p>
<p><span m="857480">All right,</span> <span m="857720">the</span> <span m="857820">proof,</span> <span m="858240">by</span> <span m="858420">PowerPoint.</span> <span m="859110">Right</span> <span m="859300">away</span> <span m="859570">you</span> <span m="859670">should</span> <span m="859850">be</span> <span m="860030">suspicious.</span> <span m="863300">I'm</span> <span m="863460">going</span> <span m="863560">to</span> <span m="863620">take</span> <span m="863860">two</span> <span m="864030">triangles</span> <span m="864770">with</span> <span m="864920">total</span> <span m="865310">area</span> <span m="865660">90</span> <span m="867050">and</span> <span m="867400">put</span> <span m="867560">them</span> <span m="867670">together,</span> <span m="868220">and</span> <span m="868320">then</span> <span m="868330">divide</span> <span m="868660">them</span> <span m="868790">up</span> <span m="868920">into</span> <span m="869080">four</span> <span m="869390">triangles</span> <span m="870540">with</span> <span m="870810">area</span> <span m="871190">greater</span> <span m="871720">than</span> <span m="871890">92.</span> <span m="874110">And</span> <span m="874390">by</span> <span m="874550">the</span> <span m="874660">conservation</span> <span m="875410">of</span> <span m="875490">area</span> <span m="875880">axiom,</span> <span m="876820">which</span> <span m="877020">you</span> <span m="877120">want</span> <span m="877310">to</span> <span m="877350">be</span> <span m="878050">maybe</span> <span m="878280">thinking</span> <span m="878570">about,</span> <span m="879230">this</span> <span m="879430">will</span> <span m="879550">imply</span> <span m="879910">an</span> <span m="880020">area</span> <span m="880310">of</span> <span m="880410">90</span> <span m="880830">is</span> <span m="880960">larger</span> <span m="881440">than</span> <span m="881540">an</span> <span m="881650">area</span> <span m="881900">of</span> <span m="881960">92.</span> <span m="883110">And</span> <span m="883240">therefore</span> <span m="883590">that</span> <span m="883770">90</span> <span m="884150">is</span> <span m="884300">bigger</span> <span m="884540">than</span> <span m="884700">92.</span> </p>
<p><span m="887460">All</span> <span m="887580">right,</span> <span m="887840">those</span> <span m="888130">are</span> <span m="888180">my</span> <span m="888340">triangles.</span> <span m="888900">They</span> <span m="889040">are</span> <span m="889240">right</span> <span m="889660">triangles,</span> <span m="890620">9</span> <span m="890880">by</span> <span m="891070">10.</span> <span m="893150">If</span> <span m="893310">I</span> <span m="893400">put</span> <span m="893610">them</span> <span m="893720">together,</span> <span m="894860">I</span> <span m="894970">get</span> <span m="895130">a</span> <span m="895200">9</span> <span m="895540">by</span> <span m="895830">10</span> <span m="896180">rectangle.</span> <span m="898510">Of course</span> <span m="898750">it</span> <span m="898820">has</span> <span m="899030">area</span> <span m="899300">90,</span> <span m="899710">9</span> <span m="900120">times</span> <span m="900380">10.</span> <span m="903140">Now</span> <span m="903340">I'm</span> <span m="903470">going</span> <span m="903580">to</span> <span m="903650">slide</span> <span m="904520">these</span> <span m="904730">triangles</span> <span m="906370">across</span> <span m="906950">their</span> <span m="908870">diagonal</span> <span m="910170">so</span> <span m="910440">that</span> <span m="910610">I</span> <span m="910730">get,</span> <span m="911150">instead</span> <span m="911450">of</span> <span m="911500">having</span> <span m="911790">10</span> <span m="912060">on</span> <span m="912140">the</span> <span m="912230">side,</span> <span m="912620">I'm</span> <span m="912690">going</span> <span m="912780">to</span> <span m="912830">slide</span> <span m="913210">it</span> <span m="913310">so</span> <span m="913400">I</span> <span m="913460">get</span> <span m="913610">a</span> <span m="913720">2</span> <span m="913940">and</span> <span m="914010">an 8.</span> <span m="915030">OK</span> <span m="915900">I</span> <span m="916030">had</span> <span m="916190">10</span> <span m="916380">there before,</span> <span m="916770">now</span> <span m="916970">it's</span> <span m="917100">2</span> <span m="917270">and</span> <span m="917360">8.</span> <span m="917910">And</span> <span m="918130">then</span> <span m="918320">you</span> <span m="918420">see</span> <span m="918580">I've</span> <span m="918680">got</span> <span m="918820">this--</span> <span m="918990">I'm</span> <span m="919060">going</span> <span m="919220">to</span> <span m="919340">cut</span> <span m="919610">off</span> <span m="919850">along</span> <span m="920110">the</span> <span m="920170">dotted</span> <span m="920490">line,</span> <span m="920940">and</span> <span m="921260">as</span> <span m="921440">you</span> <span m="921560">can</span> <span m="921710">see</span> <span m="921940">that</span> <span m="922170">dotted</span> <span m="922460">line,</span> <span m="923220">yeah but</span> <span m="923490">won't</span> <span m="923620">compute it</span> <span m="923940">exactly,</span> <span m="924380">but</span> <span m="924480">it's</span> <span m="924560">a</span> <span m="924610">little</span> <span m="924880">bit</span> <span m="925030">bigger</span> <span m="925340">than</span> <span m="925520">2.</span> <span m="926425">All right?</span> <span m="926870">It's</span> <span m="927080">a</span> <span m="927120">little</span> <span m="927330">bit</span> <span m="927460">longer</span> <span m="927810">than</span> <span m="927970">that</span> <span m="928250">2</span> <span m="928490">there,</span> <span m="928670">you could</span> <span m="928920">see</span> <span m="929150">that.</span> <span m="931090">All</span> <span m="931220">right,</span> <span m="931510">so I'll call it</span> <span m="932040">2</span> <span m="932180">plus,</span> <span m="932530">bigger</span> <span m="932710">than</span> <span m="932850">2.</span> </p>
<p><span m="933390">Now</span> <span m="933560">I</span> <span m="933660">slice</span> <span m="934230">off</span> <span m="934430">along</span> <span m="934690">the</span> <span m="934780">dotted</span> <span m="935100">line,</span> <span m="936970">and</span> <span m="937130">now</span> <span m="937310">I'm</span> <span m="937440">going</span> <span m="937540">to</span> <span m="937610">put</span> <span m="937980">those</span> <span m="938890">two</span> <span m="939110">triangles</span> <span m="939580">together,</span> <span m="940040">and</span> <span m="940120">I</span> <span m="940190">create</span> <span m="940440">a</span> <span m="940500">rectangle.</span> <span m="942560">Two</span> <span m="942770">by</span> <span m="942930">two--</span> <span m="943300">little</span> <span m="943450">more</span> <span m="943600">than</span> <span m="943740">two.</span> <span m="945010">Now</span> <span m="945140">the</span> <span m="945320">area</span> <span m="945580">of</span> <span m="945630">the</span> <span m="945710">little</span> <span m="945940">rectangle is</span> <span m="946560">bigger</span> <span m="946820">than</span> <span m="946970">4,</span> <span m="947420">cause</span> <span m="947550">I</span> <span m="947600">got</span> <span m="947870">2</span> <span m="948120">times</span> <span m="948400">something</span> <span m="948630">bigger</span> <span m="948810">than</span> <span m="948960">2,</span> <span m="949850">and</span> <span m="950160">you</span> <span m="950250">can</span> <span m="950380">see</span> <span m="950650">here</span> <span m="950980">I</span> <span m="951090">have</span> <span m="951570">the</span> <span m="951690">8</span> <span m="951860">left</span> <span m="952090">over,</span> <span m="952740">and</span> <span m="952920">I</span> <span m="952940">have</span> <span m="953060">9</span> <span m="953570">plus</span> <span m="953910">2</span> <span m="954050">plus,</span> <span m="954480">means</span> <span m="954740">it's</span> <span m="955390">a</span> <span m="955460">little</span> <span m="955670">bit</span> <span m="955820">bigger</span> <span m="956020">than</span> <span m="956140">11</span> <span m="956500">by</span> <span m="956680">8.</span> <span m="958120">I</span> <span m="958220">got</span> <span m="958400">area</span> <span m="958770">bigger</span> <span m="959250">than</span> <span m="959400">88,</span> <span m="960550">add</span> <span m="960730">them</span> <span m="960870">up,</span> <span m="962110">get</span> <span m="962300">area</span> <span m="963030">bigger</span> <span m="963250">than</span> <span m="963410">92.</span> <span m="964820">So</span> <span m="964950">I</span> <span m="965020">started</span> <span m="965340">with</span> <span m="965460">90</span> <span m="966280">and</span> <span m="966390">I</span> <span m="966450">created</span> <span m="966840">more</span> <span m="967090">than</span> <span m="967240">92.</span> </p>
<p><span m="970220">All right,</span> <span m="970410">what's</span> <span m="971150">the</span> <span m="971250">problem</span> <span m="971850">here?</span> <span m="973860">This</span> <span m="974050">would</span> <span m="974110">be</span> <span m="974180">good</span> <span m="974350">if</span> <span m="974460">I</span> <span m="974520">could</span> <span m="974700">do</span> <span m="974890">it. I'd</span> <span m="974990">get</span> <span m="975210">some</span> <span m="975470">gold,</span> <span m="975920">you know,</span> <span m="976320">bars</span> <span m="976630">of</span> <span m="976740">gold</span> <span m="977140">and</span> <span m="977900">cut</span> <span m="978100">them</span> <span m="978240">up,</span> <span m="978520">and</span> <span m="978780">do</span> <span m="978880">the</span> <span m="979010">game,</span> <span m="979360">and</span> <span m="979480">I</span> <span m="979550">make</span> <span m="979930">more</span> <span m="980140">gold.</span> <span m="980540">That</span> <span m="980700">would</span> <span m="980840">be</span> <span m="981610">pretty</span> <span m="981850">good</span> <span m="982300">if</span> <span m="982540">I</span> <span m="982610">could</span> <span m="982770">do</span> <span m="982920">that.</span> <span m="985340">Because</span> <span m="985510">of</span> <span m="985590">the</span> <span m="985670">conservation</span> <span m="986390">of</span> <span m="986470">area</span> <span m="986740">axiom?</span> <span m="988300">Did</span> <span m="988460">I</span> <span m="988590">assume</span> <span m="989160">something</span> <span m="989500">there</span> <span m="989650">that</span> <span m="989750">was</span> <span m="989860">too</span> <span m="990000">powerful</span> <span m="990570">that if I</span> <span m="991290">manipulate</span> <span m="992300">the</span> <span m="992500">area</span> <span m="992710">of</span> <span m="992770">rectangles</span> <span m="993320">like</span> <span m="993530">this,</span> <span m="994490">that the</span> <span m="994540">area</span> <span m="994770">needs</span> <span m="995000">to</span> <span m="995090">be</span> <span m="995350">the</span> <span m="995460">same?</span> <span m="999050">Yeah?</span> <span m="1001630">My</span> <span m="1001790">bigger</span> <span m="1002270">ones</span> <span m="1002740">aren't</span> <span m="1003120">closed.</span> <span m="1004170">Well,</span> <span m="1005630">let's</span> <span m="1005930">see,</span> <span m="1006120">I</span> <span m="1006170">don't</span> <span m="1006270">know.</span> <span m="1006320">Let's</span> <span m="1006500">go</span> <span m="1006590">back</span> <span m="1006800">and</span> <span m="1006910">see</span> <span m="1007040">I</span> <span m="1007140">made</span> <span m="1007360">those</span> <span m="1007550">bigger</span> <span m="1007800">ones.</span> <span m="1010230">All</span> <span m="1010320">right,</span> <span m="1010490">I</span> <span m="1010540">got</span> <span m="1010700">my</span> <span m="1010800">triangles,</span> <span m="1011570">right?</span> <span m="1012700">I'm</span> <span m="1012840">just</span> <span m="1013060">chopping</span> <span m="1013420">along</span> <span m="1013660">the</span> <span m="1013750">line</span> <span m="1014160">there,</span> <span m="1014450">and</span> <span m="1014510">I</span> <span m="1014560">got</span> <span m="1015030">2</span> <span m="1015220">plus</span> <span m="1015510">and</span> <span m="1015800">9.</span> <span m="1019210">Those</span> <span m="1019430">are</span> <span m="1019510">rectangles.</span> <span m="1021471">They</span> <span m="1021860">look</span> <span m="1022070">like</span> <span m="1022230">rectangles.</span> <span m="1025670">Yeah.</span> </p>
<p><span m="1026541">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="1029319">PROFESSOR: Well</span> <span m="1029579">it looks</span> <span m="1029819">bigger</span> <span m="1030050">than</span> <span m="1030220">2,</span> <span m="1031930">doesn't</span> <span m="1032375">it?</span> <span m="1035050">Yeah,</span> <span m="1035420">2</span> <span m="1035609">pluses.</span> <span m="1036670">Well</span> <span m="1037079">bigger</span> <span m="1037380">than</span> <span m="1037520">2</span> <span m="1037690">is</span> <span m="1037790">bigger</span> <span m="1037950">than</span> <span m="1038079">2,</span> <span m="1038260">but,</span> <span m="1038880">yeah,</span> <span m="1039099">maybe</span> <span m="1039430">that</span> <span m="1039470">should</span> <span m="1039650">have</span> <span m="1039740">been</span> <span m="1039880">a</span> <span m="1039970">2</span> <span m="1040140">minus.</span> <span m="1041329">Would</span> <span m="1041460">that</span> <span m="1041630">change</span> <span m="1041890">anything?</span> <span m="1044190">Yeah,</span> <span m="1044619">if</span> <span m="1044750">that</span> <span m="1044900">had been</span> <span m="1044960">a 2</span> <span m="1045390">minus,</span> <span m="1047030">then</span> <span m="1047200">we'd</span> <span m="1047319">have</span> <span m="1048260">area</span> <span m="1049810">less</span> <span m="1050060">than</span> <span m="1050190">4,</span> <span m="1050990">area</span> <span m="1051300">less</span> <span m="1051570">than</span> <span m="1051670">88,</span> <span m="1052170">and</span> <span m="1052250">then</span> <span m="1052500">90</span> <span m="1052860">would</span> <span m="1053010">be</span> <span m="1053230">less</span> <span m="1053480">than</span> <span m="1053590">92.</span> <span m="1054420">That</span> <span m="1054580">would</span> <span m="1054670">be</span> <span m="1054790">OK.</span> <span m="1055780">But</span> <span m="1055960">it</span> <span m="1056030">sure</span> <span m="1056350">as heck</span> <span m="1056690">looks</span> <span m="1057040">bigger</span> <span m="1057320">than--</span> <span m="1058330">the</span> <span m="1058450">2</span> <span m="1058620">plus</span> <span m="1058930">looks</span> <span m="1059120">bigger</span> <span m="1059290">than</span> <span m="1059370">the</span> <span m="1059480">2</span> <span m="1059690">doesn't</span> <span m="1059920">it?</span> <span m="1060090">I</span> <span m="1060150">don't</span> <span m="1060250">know, do</span> <span m="1060380">they</span> <span m="1060540">got</span> <span m="1060880">distortion</span> <span m="1061370">out</span> <span m="1061490">there?</span> <span m="1063240">I</span> <span m="1063360">guarantee</span> <span m="1063880">you,</span> <span m="1063980">if</span> <span m="1064100">I</span> <span m="1064180">measured</span> <span m="1064600">on</span> <span m="1064690">my</span> <span m="1064820">screen,</span> <span m="1065590">that</span> <span m="1065800">2</span> <span m="1065930">plus</span> <span m="1066240">is</span> <span m="1066360">bigger</span> <span m="1066560">than</span> <span m="1066880">the</span> <span m="1066970">length</span> <span m="1067190">of</span> <span m="1067250">the</span> <span m="1067340">tube.</span> <span m="1068890">I</span> <span m="1069010">guarantee</span> <span m="1069440">you.</span> <span m="1069530">We can</span> <span m="1070280">to</span> <span m="1070360">take</span> <span m="1070520">a</span> <span m="1070940">ruler</span> <span m="1071750">on</span> <span m="1072000">my</span> <span m="1072150">screen,</span> <span m="1072930">and</span> <span m="1073140">the</span> <span m="1073230">2</span> <span m="1073390">plus</span> <span m="1073710">will</span> <span m="1073800">be</span> <span m="1073910">longer</span> <span m="1074290">on</span> <span m="1074400">the</span> <span m="1074550">ruler</span> <span m="1074790">than</span> <span m="1074910">the</span> <span m="1075010">2.</span> <span m="1076510">That</span> <span m="1076710">I</span> <span m="1076760">guarantee</span> <span m="1077230">you,</span> <span m="1077380">but</span> <span m="1077530">you're</span> <span m="1077720">on</span> <span m="1077970">the</span> <span m="1078060">right</span> <span m="1078260">track.</span> <span m="1081090">Yeah</span> </p>
<p><span m="1084700">Good,</span> <span m="1086220">all right,</span> <span m="1086850">now</span> <span m="1087010">how</span> <span m="1087200">did</span> <span m="1087350">you--</span> <span m="1088200">but</span> <span m="1088660">it is</span> <span m="1088810">true,</span> <span m="1089200">if I</span> <span m="1089290">measured</span> <span m="1089730">on</span> <span m="1089830">my</span> <span m="1089970">screen,</span> <span m="1090310">it's</span> <span m="1090440">bigger.</span> <span m="1091100">Yeah.</span> </p>
<p><span m="1094500">AUDIENCE: Are the</span> <span m="1094650">triangles,</span> <span m="1094870">like,</span> <span m="1095130">drawn to</span> <span m="1095550">scale?</span> <span m="1095970">It's like,</span> <span m="1096390">on the</span> <span m="1096810">SAT</span> <span m="1097233">they always</span> <span m="1097656">say,</span> <span m="1098079">not to scale,</span> <span m="1098502">right?</span> </p>
<p><span m="1099350">PROFESSOR: Yeah.</span> <span m="1100720">Yeah,</span> <span m="1100910">they're</span> <span m="1101040">not</span> <span m="1101240">drawn</span> <span m="1101510">to</span> <span m="1101590">scale.</span> <span m="1102780">In</span> <span m="1102800">fact,</span> <span m="1103220">if</span> <span m="1103320">I</span> <span m="1103420">go</span> <span m="1103450">back</span> <span m="1103670">to</span> <span m="1103760">the</span> <span m="1103830">beginning,</span> <span m="1107200">it</span> <span m="1107360">says</span> <span m="1107590">9</span> <span m="1107880">by</span> <span m="1108030">10.</span> <span m="1109850">But</span> <span m="1109990">in</span> <span m="1110070">fact,</span> <span m="1110460">if</span> <span m="1110550">I</span> <span m="1110640">measure</span> <span m="1110930">the</span> <span m="1111020">nine,</span> <span m="1111350">it's</span> <span m="1111510">bigger</span> <span m="1111740">than</span> <span m="1111880">the</span> <span m="1111990">10.</span> <span m="1114080">All</span> <span m="1114180">right?</span> <span m="1115690">So</span> <span m="1116990">this</span> <span m="1117230">is</span> <span m="1117370">one</span> <span m="1117680">of</span> <span m="1117740">the</span> <span m="1117810">big</span> <span m="1118410">problems</span> <span m="1118990">of</span> <span m="1119290">proofs</span> <span m="1119580">by</span> <span m="1119710">picture.</span> <span m="1121040">Because</span> <span m="1121230">look</span> <span m="1121510">at</span> <span m="1121600">it,</span> <span m="1121870">you're</span> <span m="1122010">not</span> <span m="1122040">thinking</span> <span m="1122370">which</span> <span m="1122580">is</span> <span m="1122690">bigger,</span> <span m="1122960">9</span> <span m="1123200">or</span> <span m="1123280">10</span> <span m="1123500">up</span> <span m="1123590">there?</span> <span m="1124030">10's</span> <span m="1124310">bigger</span> <span m="1124670">even</span> <span m="1124850">though</span> <span m="1124960">it's</span> <span m="1125300">not, it's</span> <span m="1125640">shorter.</span> <span m="1126670">But</span> <span m="1126810">when</span> <span m="1126880">you</span> <span m="1127150">get</span> <span m="1127350">down into</span> <span m="1127480">the proof,</span> <span m="1127740">well,</span> <span m="1128000">clearly</span> <span m="1128430">the</span> <span m="1128530">2</span> <span m="1128710">plus</span> <span m="1129110">was</span> <span m="1129270">bigger</span> <span m="1129460">than</span> <span m="1129580">the</span> <span m="1129680">2,</span> <span m="1129860">because</span> <span m="1130040">it</span> <span m="1130140">looks</span> <span m="1130350">that</span> <span m="1130510">way.</span> <span m="1130640">It</span> <span m="1130730">is,</span> <span m="1131020">on</span> <span m="1131170">the</span> <span m="1131250">paper.</span> <span m="1132620">All</span> <span m="1132740">right?</span> <span m="1132980">In</span> <span m="1133080">fact</span> <span m="1133420">if you</span> <span m="1133530">go on a</span> <span m="1133660">computer,</span> <span m="1134060">you're</span> <span m="1134160">probably</span> <span m="1134290">right,</span> <span m="1134470">it's</span> <span m="1134570">1.8,</span> <span m="1135230">not</span> <span m="1136000">2</span> <span m="1136170">plus.</span> </p>
<p><span m="1137410">But</span> <span m="1137520">that's</span> <span m="1137730">how</span> <span m="1137850">the</span> <span m="1138000">error</span> <span m="1138315">crept in</span> <span m="1138630">and</span> <span m="1138810">how it</span> <span m="1139060">was</span> <span m="1139230">drawn,</span> <span m="1140940">and</span> <span m="1141080">then</span> <span m="1141180">you're</span> <span m="1141300">going</span> <span m="1141530">along</span> <span m="1141750">with</span> <span m="1141880">a</span> <span m="1141920">proof,</span> <span m="1142800">and</span> <span m="1142930">everything</span> <span m="1143250">else</span> <span m="1143450">is</span> <span m="1143550">just</span> <span m="1143800">fine.</span> <span m="1144040">So</span> <span m="1144160">the</span> <span m="1144500">mistake</span> <span m="1145040">is</span> <span m="1145160">right</span> <span m="1145430">up</span> <span m="1145550">front.</span> <span m="1145840">I</span> <span m="1145880">drew</span> <span m="1146130">it</span> <span m="1146210">wrong.</span> <span m="1147680">OK.</span> <span m="1149320">And</span> <span m="1149470">this</span> <span m="1149670">happens,</span> <span m="1150080">you're</span> <span m="1150210">doing</span> <span m="1150440">graphs--</span> <span m="1150960">we'll</span> <span m="1151140">talk about</span> <span m="1151190">graphs</span> <span m="1151480">here in</span> <span m="1151620">a</span> <span m="1151650">couple</span> <span m="1151870">weeks--</span> <span m="1152170">over</span> <span m="1152480">and</span> <span m="1152540">over</span> <span m="1152860">again,</span> <span m="1153520">people</span> <span m="1153810">do</span> <span m="1153970">this.</span> <span m="1154640">They</span> <span m="1154780">draw</span> <span m="1154990">it,</span> <span m="1155090">it looks</span> <span m="1155350">like</span> <span m="1155520">this,</span> <span m="1155820">you</span> <span m="1155970">accept</span> <span m="1156340">that</span> <span m="1156560">and</span> <span m="1156650">then</span> <span m="1156770">you're</span> <span m="1156860">dead.</span> <span m="1157620">Everything</span> <span m="1157950">else,</span> <span m="1158900">there's</span> <span m="1158970">no</span> <span m="1159280">hope.</span> <span m="1160190">Everybody</span> <span m="1160750">clear</span> <span m="1161020">what</span> <span m="1161170">went</span> <span m="1161290">wrong</span> <span m="1161550">in</span> <span m="1161620">this</span> <span m="1161760">picture?</span> <span m="1163080">How</span> <span m="1163280">we</span> <span m="1163350">got</span> <span m="1163840">off</span> <span m="1163980">track?</span> <span m="1165630">This</span> <span m="1165925">is</span> <span m="1166220">sort</span> <span m="1166340">of</span> <span m="1166410">a</span> <span m="1166440">nasty</span> <span m="1166750">one.</span> </p>
<p><span m="1169280">OK.</span> <span m="1172120">Now</span> <span m="1173190">one</span> <span m="1173440">of</span> <span m="1173490">the</span> <span m="1173570">nice</span> <span m="1173850">things</span> <span m="1174110">about</span> <span m="1174360">proofs</span> <span m="1174840">is</span> <span m="1174980">that</span> <span m="1175060">when</span> <span m="1175180">there</span> <span m="1175360">is</span> <span m="1175510">a</span> <span m="1175560">bug,</span> <span m="1175880">if</span> <span m="1175960">you</span> <span m="1176080">really</span> <span m="1176360">write</span> <span m="1176610">it</span> <span m="1176710">out</span> <span m="1176880">step</span> <span m="1177100">by</span> <span m="1177230">step,</span> <span m="1178160">and</span> <span m="1178320">there's</span> <span m="1178450">a</span> <span m="1178510">bug,</span> <span m="1178775">you can</span> <span m="1179040">go</span> <span m="1179140">back</span> <span m="1179340">and</span> <span m="1179470">find</span> <span m="1179780">it.</span> <span m="1180620">And</span> <span m="1180820">so</span> <span m="1180960">this,</span> <span m="1181370">I</span> <span m="1181420">didn't</span> <span m="1181670">sort</span> <span m="1181860">of</span> <span m="1181980">really</span> <span m="1182240">write</span> <span m="1182470">it</span> <span m="1182560">out</span> <span m="1182740">very</span> <span m="1182900">well</span> <span m="1183110">step</span> <span m="1183330">by</span> <span m="1183450">step,</span> <span m="1183770">and it</span> <span m="1183930">was</span> <span m="1184480">harder</span> <span m="1185410">to</span> <span m="1185480">find.</span> <span m="1186740">Now,</span> <span m="1187680">all</span> <span m="1187770">right,</span> <span m="1187960">so</span> <span m="1188040">we</span> <span m="1188130">can</span> <span m="1188230">pull</span> <span m="1188390">the</span> <span m="1188480">screens</span> <span m="1188830">up.</span> <span m="1192543">Thanks.</span> <span m="1194460">OK,</span> <span m="1194820">so</span> <span m="1196170">for</span> <span m="1196290">the</span> <span m="1196380">rest</span> <span m="1196610">of</span> <span m="1196690">today</span> <span m="1197200">and</span> <span m="1197340">the</span> <span m="1197430">rest</span> <span m="1197660">of</span> <span m="1197710">this</span> <span m="1197880">week,</span> <span m="1198400">we're</span> <span m="1198540">going</span> <span m="1198630">to</span> <span m="1198690">talk</span> <span m="1198950">about</span> <span m="1199180">a</span> <span m="1199220">different</span> <span m="1199510">kind</span> <span m="1199670">of</span> <span m="1199740">proof</span> <span m="1200040">technique,</span> <span m="1200930">which</span> <span m="1201240">is</span> <span m="1201570">induction.</span> <span m="1203600">Now,</span> <span m="1203950">induction</span> <span m="1204470">is</span> <span m="1204570">by</span> <span m="1204710">far</span> <span m="1205230">the</span> <span m="1205330">most</span> <span m="1205600">powerful</span> <span m="1206550">and</span> <span m="1206670">commonly</span> <span m="1207190">used</span> <span m="1207510">proof</span> <span m="1207720">technique</span> <span m="1208790">in</span> <span m="1208900">computer</span> <span m="1209260">science.</span> <span m="1210350">If</span> <span m="1210510">there's</span> <span m="1210710">one</span> <span m="1211000">thing</span> <span m="1211220">you</span> <span m="1211410">should</span> <span m="1211620">know</span> <span m="1211880">by</span> <span m="1211990">the</span> <span m="1212110">time</span> <span m="1212430">you're</span> <span m="1212550">done</span> <span m="1212800">with</span> <span m="1212910">this</span> <span m="1213090">class,</span> <span m="1214640">it's</span> <span m="1214820">how</span> <span m="1215040">to</span> <span m="1215130">do</span> <span m="1215250">a</span> <span m="1215330">proof</span> <span m="1215560">by</span> <span m="1215740">induction.</span> <span m="1216950">In fact,</span> <span m="1217140">if</span> <span m="1217200">there's</span> <span m="1217340">one</span> <span m="1217530">thing</span> <span m="1217660">you</span> <span m="1217790">will</span> <span m="1218080">know</span> <span m="1218700">by the</span> <span m="1218880">time we're</span> <span m="1219270">done with</span> <span m="1219410">this</span> <span m="1219600">class,</span> <span m="1220280">is</span> <span m="1220420">how</span> <span m="1220550">to</span> <span m="1220640">do</span> <span m="1220760">a</span> <span m="1220840">proof</span> <span m="1221080">by</span> <span m="1221250">induction.</span> </p>
<p><span m="1222020">And</span> <span m="1222160">in</span> <span m="1222230">some</span> <span m="1222500">sense, when</span> <span m="1222840">we</span> <span m="1223110">get</span> <span m="1223290">to</span> <span m="1223370">grading</span> <span m="1223750">the</span> <span m="1223830">final,</span> <span m="1225010">how</span> <span m="1225240">we</span> <span m="1225390">measure</span> <span m="1225760">ourselves</span> <span m="1226070">as</span> <span m="1226380">instructors,</span> <span m="1227020">the</span> <span m="1227100">first</span> <span m="1227510">test</span> <span m="1227900">is,</span> <span m="1228830">can</span> <span m="1229160">you</span> <span m="1229260">do</span> <span m="1229380">the</span> <span m="1229520">proof</span> <span m="1229770">by</span> <span m="1229930">induction</span> <span m="1230320">question,</span> <span m="1230900">and</span> <span m="1231080">do</span> <span m="1231200">a</span> <span m="1231270">good</span> <span m="1231450">proof</span> <span m="1231710">there?</span> <span m="1232250">You</span> <span m="1232410">will</span> <span m="1232550">see</span> <span m="1232780">it</span> <span m="1232860">on</span> <span m="1233780">the</span> <span m="1233870">midterm</span> <span m="1234450">and</span> <span m="1234640">the</span> <span m="1234720">final,</span> <span m="1235090">probably</span> <span m="1235480">in</span> <span m="1235570">multiple</span> <span m="1236000">instances.</span> <span m="1238410">Now</span> <span m="1238930">just</span> <span m="1239180">to</span> <span m="1239240">make</span> <span m="1239390">sure</span> <span m="1239560">you</span> <span m="1239660">become</span> <span m="1240010">intimately</span> <span m="1240540">familiar</span> <span m="1240960">with</span> <span m="1241110">induction,</span> <span m="1242350">we're</span> <span m="1242570">going</span> <span m="1242700">to</span> <span m="1242820">do</span> <span m="1243490">over</span> <span m="1243730">a</span> <span m="1243780">dozen</span> <span m="1244140">proofs</span> <span m="1244480">with</span> <span m="1244610">induction</span> <span m="1245000">in</span> <span m="1245080">class,</span> <span m="1245540">and</span> <span m="1245870">many</span> <span m="1246050">more in</span> <span m="1246270">homework</span> <span m="1246720">over</span> <span m="1246860">the</span> <span m="1246930">next</span> <span m="1247440">five</span> <span m="1247660">to</span> <span m="1247740">six</span> <span m="1248000">weeks.</span> <span m="1249610">You'll</span> <span m="1249740">probably</span> <span m="1250040">be</span> <span m="1250160">dreaming</span> <span m="1250950">about</span> <span m="1251240">induction,</span> <span m="1251670">if</span> <span m="1251800">we're</span> <span m="1251960">successful</span> <span m="1252490">here.</span> <span m="1253940">Soon.</span> </p>
<p><span m="1255130">The</span> <span m="1255240">good</span> <span m="1255460">news</span> <span m="1255700">is</span> <span m="1256350">that</span> <span m="1256480">induction</span> <span m="1256970">is</span> <span m="1257090">very</span> <span m="1257550">easy.</span> <span m="1258210">Once</span> <span m="1258480">you</span> <span m="1258550">get</span> <span m="1258700">your</span> <span m="1258800">mind</span> <span m="1259100">around</span> <span m="1259350">it,</span> <span m="1259440">get</span> <span m="1259590">some</span> <span m="1259710">practice,</span> <span m="1260150">it</span> <span m="1260520">really</span> <span m="1260910">is</span> <span m="1261040">not</span> <span m="1261280">a</span> <span m="1261340">hard</span> <span m="1261630">thing.</span> <span m="1262910">In</span> <span m="1263050">fact,</span> <span m="1263410">induction</span> <span m="1263910">is</span> <span m="1264020">really</span> <span m="1264390">just</span> <span m="1264990">an</span> <span m="1265130">axiom.</span> <span m="1268290">So</span> <span m="1268450">let</span> <span m="1268540">me</span> <span m="1268650">state</span> <span m="1268920">it.</span> <span m="1276950">Let</span> <span m="1277280">P(n)</span> <span m="1279180">be a</span> <span m="1279450">predicate.</span> <span m="1286740">If</span> <span m="1287410">P(0)</span> <span m="1288721">is</span> <span m="1289160">true,</span> <span m="1293110">and</span> <span m="1296100">for</span> <span m="1296390">all</span> <span m="1297500">natural</span> <span m="1297910">numbers</span> <span m="1298340">n,</span> <span m="1303670">p</span> <span m="1303960">of</span> <span m="1304040">n</span> <span m="1304940">implies</span> <span m="1305840">p</span> <span m="1306140">of</span> <span m="1306220">n</span> <span m="1306390">plus</span> <span m="1306680">1</span> <span m="1308770">is</span> <span m="1308950">true.</span> <span m="1311140">So</span> <span m="1311180">here</span> <span m="1311390">I'm</span> <span m="1311500">saying</span> <span m="1311920">that</span> <span m="1312060">this</span> <span m="1312590">is</span> <span m="1312750">true</span> <span m="1313140">for</span> <span m="1313360">all</span> <span m="1313570">n.</span> <span m="1319770">OK.</span> <span m="1322620">Then</span> <span m="1324270">for</span> <span m="1326380">all</span> <span m="1326640">n,</span> <span m="1328980">natural</span> <span m="1329290">numbers,</span> <span m="1331010">P</span> <span m="1331220">of</span> <span m="1331300">n</span> <span m="1331450">is</span> <span m="1331540">true.</span> <span m="1336170">I</span> <span m="1336370">had</span> <span m="1336450">another</span> <span m="1336670">way</span> <span m="1336840">of</span> <span m="1336930">saying</span> <span m="1337330">this</span> <span m="1337530">without</span> <span m="1338090">the</span> <span m="1338220">for</span> <span m="1338485">alls</span> <span m="1338750">there,</span> <span m="1339970">is</span> <span m="1340220">that</span> <span m="1341510">if</span> <span m="1343030">P(0)</span> <span m="1343670">is</span> <span m="1343790">true,</span> <span m="1345090">if</span> <span m="1345260">P(0)</span> <span m="1346370">implies</span> <span m="1347050">P(1)</span> <span m="1347600">is</span> <span m="1347710">true,</span> <span m="1349130">if</span> <span m="1350380">P(1)</span> <span m="1351690">implies</span> <span m="1352270">P(2)</span> <span m="1352720">is</span> <span m="1352850">true,</span> <span m="1354030">and</span> <span m="1354330">so</span> <span m="1354600">on,</span> <span m="1354950">forever,</span> <span m="1357140">then</span> <span m="1358070">p</span> <span m="1358270">of</span> <span m="1358330">n</span> <span m="1358480">is</span> <span m="1358590">true</span> <span m="1358790">for</span> <span m="1358960">all</span> <span m="1359110">that.</span> <span m="1360240">So</span> <span m="1360470">then,</span> <span m="1362630">P(0),</span> <span m="1364320">P(1),</span> <span m="1366010">P(2),</span> <span m="1367690">forever,</span> <span m="1368550">are</span> <span m="1368680">true.</span> </p>
<p><span m="1377760">OK.</span> <span m="1379900">Now</span> <span m="1380300">you</span> <span m="1380440">can</span> <span m="1380550">sort</span> <span m="1380820">of</span> <span m="1380900">see</span> <span m="1381130">why</span> <span m="1381290">this</span> <span m="1381490">is</span> <span m="1381610">a</span> <span m="1381720">reasonable</span> <span m="1382630">axiom,</span> <span m="1384600">because</span> <span m="1385110">if</span> <span m="1385260">P(0)</span> <span m="1385860">is</span> <span m="1385960">true,</span> <span m="1387380">and</span> <span m="1388090">P(0)</span> <span m="1388490">implies</span> <span m="1388870">P(1),</span> <span m="1390240">then</span> <span m="1390440">by</span> <span m="1390570">that</span> <span m="1390770">modus</span> <span m="1391080">ponens</span> <span m="1391510">thing</span> <span m="1391710">or</span> <span m="1391780">one</span> <span m="1391920">of</span> <span m="1391960">the</span> <span m="1392020">logical</span> <span m="1392420">deductions,</span> <span m="1393640">we</span> <span m="1393740">know</span> <span m="1393880">P(1)</span> <span m="1394220">is</span> <span m="1394300">true.</span> <span m="1396000">And</span> <span m="1396310">if</span> <span m="1396850">P(1)</span> <span m="1397230">is</span> <span m="1397330">true,</span> <span m="1398400">and</span> <span m="1398580">P(1)</span> <span m="1398750">implies</span> <span m="1399230">P(2)</span> <span m="1399590">is</span> <span m="1399700">true,</span> <span m="1400120">then</span> <span m="1400220">we</span> <span m="1400310">know</span> <span m="1400560">by</span> <span m="1400670">the</span> <span m="1400790">same</span> <span m="1401060">reasoning,</span> <span m="1401770">P(2)</span> <span m="1402140">is</span> <span m="1402250">true,</span> <span m="1404020">and</span> <span m="1404150">so</span> <span m="1404480">on,</span> <span m="1404710">forever.</span> <span m="1405770">Now</span> <span m="1405890">the</span> <span m="1406000">reason</span> <span m="1406300">it</span> <span m="1406350">becomes</span> <span m="1406660">an</span> <span m="1406760">axiom</span> <span m="1407310">is</span> <span m="1407410">that</span> <span m="1407930">and</span> <span m="1408110">so</span> <span m="1408360">on</span> <span m="1408530">forever</span> <span m="1408870">bit</span> <span m="1408980">is</span> <span m="1409090">part</span> <span m="1409310">of</span> <span m="1409410">it.</span> <span m="1410200">That's</span> <span m="1410430">why</span> <span m="1410530">we</span> <span m="1410650">need</span> <span m="1410810">it as</span> <span m="1410980">an</span> <span m="1411070">axiom.</span> <span m="1413370">You</span> <span m="1413510">could</span> <span m="1413650">sort</span> <span m="1413820">of</span> <span m="1413890">view</span> <span m="1414150">this</span> <span m="1414440">as</span> <span m="1414720">a</span> <span m="1415110">series</span> <span m="1415420">of</span> <span m="1415540">dominoes.</span> <span m="1417652">You know,</span> <span m="1418020">I</span> <span m="1418140">got</span> <span m="1418320">a</span> <span m="1418390">domino</span> <span m="1418970">for</span> <span m="1419190">each</span> <span m="1420030">n,</span> <span m="1437300">and</span> <span m="1438230">each</span> <span m="1438630">domino</span> <span m="1439140">knocks</span> <span m="1439520">over</span> <span m="1439710">the</span> <span m="1439820">next</span> <span m="1440085">in</span> <span m="1440350">terms</span> <span m="1440750">of</span> <span m="1440850">truth,</span> <span m="1441710">knocking</span> <span m="1442040">over</span> <span m="1442260">corresponds</span> <span m="1442840">to</span> <span m="1442940">the</span> <span m="1443040">truth</span> <span m="1443200">here.</span> <span m="1444230">And</span> <span m="1444460">if</span> <span m="1444570">I</span> <span m="1445210">know</span> <span m="1445390">P(0)</span> <span m="1445860">is</span> <span m="1445950">true,</span> <span m="1446190">that</span> <span m="1446330">means</span> <span m="1446520">I</span> <span m="1446580">knock</span> <span m="1446870">over</span> <span m="1447040">P(0),</span> <span m="1448400">P(0)</span> <span m="1448800">implies</span> <span m="1449120">P(1)</span> <span m="1449470">means</span> <span m="1449680">P(1)</span> <span m="1449830">one</span> <span m="1450080">goes</span> <span m="1450270">down.</span> <span m="1451210">P(1)</span> <span m="1451300">implies</span> <span m="1451740">P(2)</span> <span m="1452060">means</span> <span m="1452360">P(2)</span> <span m="1452540">goes</span> <span m="1452700">down,</span> <span m="1453110">and</span> <span m="1453300">so</span> <span m="1453460">forth.</span> </p>
<p><span m="1455070">Pretty</span> <span m="1455920">basic.</span> <span m="1459380">Raise</span> <span m="1459610">your</span> <span m="1459720">hand</span> <span m="1459980">if</span> <span m="1460050">you</span> <span m="1460150">think</span> <span m="1460360">you</span> <span m="1460480">don't</span> <span m="1460720">have</span> <span m="1460900">a</span> <span m="1460940">lot</span> <span m="1461120">of</span> <span m="1461190">experience</span> <span m="1461660">with</span> <span m="1461780">induction.</span> <span m="1463745">All right,</span> <span m="1464210">yeah,</span> <span m="1464350">that's</span> <span m="1465230">pretty</span> <span m="1465790">typical.</span> <span m="1466200">About</span> <span m="1466400">a</span> <span m="1466530">third</span> <span m="1466630">to a</span> <span m="1466680">half</span> <span m="1466900">of</span> <span m="1466980">you.</span> <span m="1467560">So</span> <span m="1467720">we're</span> <span m="1468140">going</span> <span m="1468270">to</span> <span m="1468340">change</span> <span m="1468640">that.</span> <span m="1470640">Let's</span> <span m="1470860">do</span> <span m="1470970">a</span> <span m="1471030">simple</span> <span m="1471330">proof</span> <span m="1471680">using</span> <span m="1471970">induction.</span> <span m="1476580">So</span> <span m="1476750">let's</span> <span m="1477190">prove</span> <span m="1477600">that</span> <span m="1477700">for all</span> <span m="1478250">n</span> <span m="1479280">bigger</span> <span m="1479500">than</span> <span m="1479680">or</span> <span m="1479750">equal</span> <span m="1479940">to</span> <span m="1480000">0,</span> <span m="1480600">again</span> <span m="1480890">natural</span> <span m="1481230">numbers,</span> <span m="1482160">that</span> <span m="1482300">1,</span> <span m="1482660">plus</span> <span m="1482950">2,</span> <span m="1483410">plus</span> <span m="1483700">3,</span> <span m="1485820">plus</span> <span m="1486890">n</span> <span m="1488140">equals</span> <span m="1489360">n</span> <span m="1489730">times</span> <span m="1490000">n</span> <span m="1490140">plus</span> <span m="1490380">1</span> <span m="1491340">over</span> <span m="1491580">2.</span> <span m="1493980">This</span> <span m="1494140">is</span> <span m="1494240">actually a</span> <span m="1494590">useful</span> <span m="1494900">thing</span> <span m="1495090">to</span> <span m="1495160">remember.</span> <span m="1495510">We're</span> <span m="1495640">going</span> <span m="1495750">to</span> <span m="1495790">use</span> <span m="1496080">this</span> <span m="1496410">all</span> <span m="1496745">term, this</span> <span m="1497080">identity.</span> </p>
<p><span m="1498820">Now</span> <span m="1500040">the</span> <span m="1500140">first</span> <span m="1500430">thing,</span> <span m="1500660">before</span> <span m="1500900">we</span> <span m="1501010">prove</span> <span m="1501350">it,</span> <span m="1501740">I</span> <span m="1501810">want</span> <span m="1501960">to</span> <span m="1502000">make</span> <span m="1502130">sure</span> <span m="1502320">we</span> <span m="1502450">understand</span> <span m="1502950">this</span> <span m="1503780">dot,</span> <span m="1504080">dot,</span> <span m="1504340">dot,</span> <span m="1504710">notation.</span> <span m="1505740">Because</span> <span m="1505890">it</span> <span m="1505980">is</span> <span m="1506110">the</span> <span m="1506210">source</span> <span m="1506630">of</span> <span m="1506720">a</span> <span m="1506790">lot</span> <span m="1507100">of</span> <span m="1507190">errors.</span> <span m="1508690">What</span> <span m="1508890">it</span> <span m="1509040">means</span> <span m="1509800">is</span> <span m="1510080">that</span> <span m="1510810">you</span> <span m="1510950">need</span> <span m="1511120">to</span> <span m="1511190">fill</span> <span m="1511600">in</span> <span m="1511830">the</span> <span m="1511920">pattern</span> <span m="1512370">here,</span> <span m="1514180">which</span> <span m="1514550">is</span> <span m="1514820">vague.</span> <span m="1516500">What</span> <span m="1516630">it</span> <span m="1516700">means</span> <span m="1516940">in</span> <span m="1516990">this</span> <span m="1517210">case,</span> <span m="1517450">you</span> <span m="1517670">fill</span> <span m="1517930">in</span> <span m="1518030">four,</span> <span m="1518370">five,</span> <span m="1518700">six</span> <span m="1519220">all</span> <span m="1519380">the</span> <span m="1519440">way</span> <span m="1519540">up</span> <span m="1519650">to</span> <span m="1519970">n</span> <span m="1520150">minus</span> <span m="1520510">1,</span> <span m="1521160">and then</span> <span m="1521460">n.</span> <span m="1522020">That's</span> <span m="1522240">what</span> <span m="1522350">it</span> <span m="1522430">means.</span> <span m="1522740">Figure</span> <span m="1523060">out</span> <span m="1523200">the</span> <span m="1523290">pattern</span> <span m="1523690">and</span> <span m="1523810">fill</span> <span m="1524120">it in.</span> <span m="1525020">Pretty</span> <span m="1525230">risky</span> <span m="1525880">thing.</span> <span m="1527340">Now</span> <span m="1527480">in</span> <span m="1527560">this</span> <span m="1527750">case,</span> <span m="1528050">because</span> <span m="1528390">that's</span> <span m="1528610">so</span> <span m="1528770">vague,</span> <span m="1529490">there's</span> <span m="1529680">other</span> <span m="1529990">terminology</span> <span m="1530640">we</span> <span m="1530770">use</span> <span m="1531050">for</span> <span m="1531140">this.</span> <span m="1532320">For</span> <span m="1532420">example,</span> <span m="1533030">we</span> <span m="1533150">would</span> <span m="1533290">use</span> <span m="1535150">a</span> <span m="1535230">big</span> <span m="1535710">sigma,</span> <span m="1536200">capital</span> <span m="1536560">sigma</span> <span m="1537720">i</span> <span m="1538060">equals</span> <span m="1538470">0</span> <span m="1538780">to</span> <span m="1539070">n</span> <span m="1539550">of</span> <span m="1539890">i.</span> <span m="1540250">That</span> <span m="1540310">means</span> <span m="1540580">the</span> <span m="1540700">sum</span> <span m="1542080">of</span> <span m="1542290">i,</span> <span m="1543170">where i</span> <span m="1543610">is</span> <span m="1543750">the</span> <span m="1543920">integers</span> <span m="1544150">from</span> <span m="1544490">0</span> <span m="1544940">to</span> <span m="1545090">n</span> <span m="1545310">inclusive.</span> <span m="1547030">Actually,</span> <span m="1547330">let me</span> <span m="1547440">put</span> <span m="1547620">1</span> <span m="1547850">here.</span> </p>
<p><span m="1551470">Another</span> <span m="1551640">way</span> <span m="1551750">to</span> <span m="1551860">write</span> <span m="1552120">this--</span> <span m="1553491">these</span> <span m="1553950">are</span> <span m="1554050">all</span> <span m="1554270">equivalent</span> <span m="1554620">ways</span> <span m="1554830">to</span> <span m="1554930">write</span> <span m="1555150">it--</span> <span m="1556270">is</span> <span m="1556460">you</span> <span m="1556600">could</span> <span m="1556760">put</span> <span m="1557410">1</span> <span m="1557510">less</span> <span m="1557610">than</span> <span m="1557810">or</span> <span m="1558160">equal</span> <span m="1558350">to</span> <span m="1558510">i,</span> <span m="1559030">less than or</span> <span m="1559270">equal</span> <span m="1559490">to</span> <span m="1559580">n</span> <span m="1560430">of</span> <span m="1560590">i.</span> <span m="1561005">So</span> <span m="1561420">you</span> <span m="1561480">could</span> <span m="1561640">put</span> <span m="1562430">something</span> <span m="1562790">i</span> <span m="1563010">over</span> <span m="1563200">the</span> <span m="1563330">range,</span> <span m="1563850">from</span> <span m="1564000">one to</span> <span m="1564230">n,</span> <span m="1564810">or</span> <span m="1565230">you</span> <span m="1565370">can</span> <span m="1565510">write</span> <span m="1565780">it</span> <span m="1565940">on</span> <span m="1566080">the</span> <span m="1566150">bottom</span> <span m="1566550">if</span> <span m="1566630">you</span> <span m="1566760">want.</span> <span m="1570690">So</span> <span m="1570810">these</span> <span m="1571000">are</span> <span m="1571050">four</span> <span m="1571560">different</span> <span m="1571910">ways</span> <span m="1572270">of</span> <span m="1572380">writing</span> <span m="1572730">the</span> <span m="1572820">same</span> <span m="1573160">thing,</span> <span m="1573530">the</span> <span m="1573670">sum</span> <span m="1573900">of</span> <span m="1573970">the</span> <span m="1574060">natural</span> <span m="1574480">numbers</span> <span m="1574850">from</span> <span m="1575050">1</span> <span m="1575260">to</span> <span m="1575370">n.</span> <span m="1576150">And</span> <span m="1576310">we'll</span> <span m="1576460">use</span> <span m="1576660">them</span> <span m="1576710">all</span> <span m="1577190">during</span> <span m="1577370">the</span> <span m="1577460">term.</span> <span m="1580810">All</span> <span m="1580970">right,</span> <span m="1581210">now</span> <span m="1581460">there's</span> <span m="1581620">some</span> <span m="1581740">special</span> <span m="1582210">cases</span> <span m="1582750">that</span> <span m="1583680">make</span> <span m="1583920">this</span> <span m="1585300">a</span> <span m="1585320">little</span> <span m="1585470">more</span> <span m="1585660">interesting.</span> <span m="1586430">What</span> <span m="1586670">if</span> <span m="1586790">n</span> <span m="1586980">equals</span> <span m="1587280">1?</span> </p>
<p><span m="1590520">I've</span> <span m="1590710">got</span> <span m="1590900">1</span> <span m="1591240">plus</span> <span m="1591570">2</span> <span m="1592780">plus</span> <span m="1593140">dot,</span> <span m="1593340">dot, dot,</span> <span m="1594080">plus</span> <span m="1595250">n.</span> <span m="1598490">What</span> <span m="1598620">do</span> <span m="1598660">you</span> <span m="1598750">suppose</span> <span m="1599180">it</span> <span m="1599250">equals</span> <span m="1599620">if</span> <span m="1599720">n</span> <span m="1599860">equals</span> <span m="1600100">1?</span> <span m="1603290">1,</span> <span m="1604350">because</span> <span m="1604560">we're</span> <span m="1604670">summing</span> <span m="1605160">the</span> <span m="1605260">numbers</span> <span m="1605880">from</span> <span m="1606130">1</span> <span m="1607600">to</span> <span m="1607780">1.</span> <span m="1608910">That's</span> <span m="1609170">just</span> <span m="1609780">1,</span> <span m="1610440">the</span> <span m="1610550">number</span> <span m="1610800">1.</span> <span m="1611390">There</span> <span m="1611570">is</span> <span m="1611800">no</span> <span m="1611980">2.</span> <span m="1613930">And</span> <span m="1614100">there</span> <span m="1614180">aren't</span> <span m="1614580">two</span> <span m="1614790">copies</span> <span m="1615220">of</span> <span m="1615310">1.</span> <span m="1616880">So</span> <span m="1617010">this</span> <span m="1617210">notation</span> <span m="1617700">is</span> <span m="1617800">very</span> <span m="1618310">ambiguous.</span> <span m="1618910">You'll</span> <span m="1619010">see</span> <span m="1619220">a</span> <span m="1619310">2</span> <span m="1619590">here</span> <span m="1619750">in</span> <span m="1619840">the</span> <span m="1619930">sum.</span> <span m="1620810">If</span> <span m="1620950">n</span> <span m="1621100">is</span> <span m="1621200">1,</span> <span m="1621430">you'll</span> <span m="1621530">see</span> <span m="1621690">a</span> <span m="1621750">1</span> <span m="1621980">here</span> <span m="1622300">and</span> <span m="1622420">a</span> <span m="1622480">1</span> <span m="1622710">here.</span> </p>
<p><span m="1624180">So you've</span> <span m="1624250">got</span> <span m="1624400">to</span> <span m="1624450">be</span> <span m="1624560">careful.</span> <span m="1624780">I</span> <span m="1624900">guarantee</span> <span m="1625170">you,</span> <span m="1625440">you'll</span> <span m="1625700">make</span> <span m="1625880">a</span> <span m="1625930">mistake</span> <span m="1626770">with</span> <span m="1626890">this.</span> <span m="1627080">In fact</span> <span m="1627420">I'm</span> <span m="1627480">going</span> <span m="1627570">to</span> <span m="1627610">show</span> <span m="1627810">you</span> <span m="1627950">another</span> <span m="1628190">false</span> <span m="1628420">proof</span> <span m="1628630">later</span> <span m="1628880">where</span> <span m="1628970">this</span> <span m="1629180">comes</span> <span m="1629390">into</span> <span m="1629510">play.</span> <span m="1631520">What</span> <span m="1631670">about</span> <span m="1631870">if</span> <span m="1632050">n</span> <span m="1632200">is</span> <span m="1632330">less</span> <span m="1632570">than</span> <span m="1632670">or</span> <span m="1632770">equal</span> <span m="1632820">to</span> <span m="1632880">zero?</span> <span m="1635880">What</span> <span m="1636110">is</span> <span m="1636240">it then?</span> <span m="1640950">Any</span> <span m="1641150">thoughts?</span> <span m="1645620">0.</span> </p>
<p><span m="1646680">There</span> <span m="1646930">are</span> <span m="1647300">no</span> <span m="1647950">integers</span> <span m="1648760">to</span> <span m="1648970">sum,</span> <span m="1651370">no</span> <span m="1651560">1,</span> <span m="1651900">no</span> <span m="1652030">2,</span> <span m="1652260">no</span> <span m="1652400">n,</span> <span m="1652550">because</span> <span m="1653310">you never</span> <span m="1653470">get</span> <span m="1653600">started</span> <span m="1653940">because--</span> <span m="1654270">sorry,</span> <span m="1654560">n is</span> <span m="1654810">less than</span> <span m="1654930">or equal</span> <span m="1655050">to</span> <span m="1655160">0--</span> <span m="1655650">because</span> <span m="1655860">you're</span> <span m="1656010">summing</span> <span m="1656330">from</span> <span m="1656610">1</span> <span m="1656870">to</span> <span m="1656920">0,</span> <span m="1657550">0 is</span> <span m="1657910">below.</span> <span m="1658320">You</span> <span m="1658410">never--</span> <span m="1658840">it</span> <span m="1659160">doesn't</span> <span m="1659370">include</span> <span m="1659630">anything.</span> <span m="1660710">So</span> <span m="1661040">these</span> <span m="1661310">are</span> <span m="1661350">the</span> <span m="1661460">conventions</span> <span m="1661990">to</span> <span m="1662070">keep</span> <span m="1662300">in</span> <span m="1662380">mind</span> <span m="1664181">with</span> <span m="1664610">the</span> <span m="1664850">edge</span> <span m="1665120">cases</span> <span m="1665510">here.</span> <span m="1670500">All right,</span> <span m="1670660">it's</span> <span m="1670810">easy</span> <span m="1671270">enough to</span> <span m="1671350">check</span> <span m="1671910">that</span> <span m="1672400">the</span> <span m="1672490">theorem</span> <span m="1672890">is</span> <span m="1673300">true</span> <span m="1673640">for</span> <span m="1673800">certain</span> <span m="1674040">values</span> <span m="1674350">of</span> <span m="1674440">n.</span> </p>
<p><span m="1674640">For</span> <span m="1674730">example,</span> <span m="1675735">if</span> <span m="1676040">n</span> <span m="1676190">equals</span> <span m="1676510">4,</span> <span m="1677020">I've</span> <span m="1677060">got</span> <span m="1677300">1</span> <span m="1677620">plus</span> <span m="1677930">2</span> <span m="1678450">plus</span> <span m="1678790">3</span> <span m="1679270">plus</span> <span m="1679580">4.</span> <span m="1680520">That's</span> <span m="1680860">10.</span> <span m="1682220">And</span> <span m="1682440">10</span> <span m="1682690">equals</span> <span m="1683630">4</span> <span m="1684230">times</span> <span m="1685340">5</span> <span m="1686540">over</span> <span m="1686760">2,</span> <span m="1687730">plugging</span> <span m="1688020">in</span> <span m="1688100">the</span> <span m="1688180">formula.</span> <span m="1689580">So</span> <span m="1689630">for</span> <span m="1689770">any</span> <span m="1689990">value</span> <span m="1690370">of n</span> <span m="1690610">you</span> <span m="1690710">could</span> <span m="1690860">check</span> <span m="1691120">this</span> <span m="1691290">formula is</span> <span m="1691700">true.</span> <span m="1693030">Proving</span> <span m="1693440">is</span> <span m="1693550">true</span> <span m="1693740">for</span> <span m="1693910">all</span> <span m="1694230">n.</span> <span m="1695930">It</span> <span m="1696370">takes</span> <span m="1696580">a</span> <span m="1696630">little</span> <span m="1696740">more</span> <span m="1696920">effort</span> <span m="1697240">unless</span> <span m="1697560">you</span> <span m="1697680">use</span> <span m="1698220">induction.</span> <span m="1699820">So</span> <span m="1700140">let's</span> <span m="1700360">do</span> <span m="1700490">that.</span> </p>
<p><span m="1703430">So we'll</span> <span m="1703630">prove</span> <span m="1703890">the</span> <span m="1703990">theorem.</span> <span m="1704870">Now,</span> <span m="1705020">whenever</span> <span m="1705270">you're</span> <span m="1705370">using</span> <span m="1705610">a</span> <span m="1705670">proof</span> <span m="1705890">by</span> <span m="1706050">induction,</span> <span m="1707200">first</span> <span m="1707450">thing</span> <span m="1707540">you</span> <span m="1707630">do</span> <span m="1707880">is</span> <span m="1707960">you</span> <span m="1708080">write</span> <span m="1708290">down</span> <span m="1708590">by</span> <span m="1708760">induction</span> <span m="1710480">so</span> <span m="1710620">we</span> <span m="1710730">know</span> <span m="1710890">what</span> <span m="1711000">you're</span> <span m="1711100">going</span> <span m="1711200">to</span> <span m="1711270">do.</span> <span m="1713920">And</span> <span m="1713990">the</span> <span m="1714750">next</span> <span m="1714930">thing</span> <span m="1715050">you</span> <span m="1715160">need</span> <span m="1715670">to</span> <span m="1715740">do</span> <span m="1715980">is</span> <span m="1716100">figure</span> <span m="1716410">out,</span> <span m="1717830">what's</span> <span m="1718090">your</span> <span m="1718670">predicate.</span> <span m="1720500">What's</span> <span m="1720730">your</span> <span m="1721470">inductive</span> <span m="1721930">hypothesis?</span> <span m="1723060">What's</span> <span m="1723300">p?</span> </p>
<p><span m="1726310">So</span> <span m="1727360">usually</span> <span m="1728930">p</span> <span m="1729340">will</span> <span m="1729480">be</span> <span m="1730370">the</span> <span m="1730480">thing</span> <span m="1730750">you're</span> <span m="1730870">trying</span> <span m="1731200">to</span> <span m="1731270">prove,</span> <span m="1731680">namely</span> <span m="1731980">that</span> <span m="1732100">1 plus</span> <span m="1732590">2</span> <span m="1732710">plus</span> <span m="1732960">3</span> <span m="1733320">up</span> <span m="1733420">to</span> <span m="1733510">n</span> <span m="1733820">is</span> <span m="1734440">n</span> <span m="1734640">times</span> <span m="1734880">n</span> <span m="1734980">plus</span> <span m="1735160">1</span> <span m="1735300">over</span> <span m="1735450">2.</span> <span m="1736460">And</span> <span m="1736780">you</span> <span m="1736920">state</span> <span m="1737230">that.</span> <span m="1737390">You</span> <span m="1737460">say</span> <span m="1737600">let</span> <span m="1737890">p</span> <span m="1738120">of</span> <span m="1738200">n</span> <span m="1739875">be</span> <span m="1740280">the</span> <span m="1741010">proposition,</span> <span m="1744760">the</span> <span m="1744880">predicate,</span> <span m="1746140">that</span> <span m="1748160">the</span> <span m="1748260">sum</span> <span m="1749190">i</span> <span m="1749620">equals</span> <span m="1750060">1</span> <span m="1750430">to</span> <span m="1750540">n</span> <span m="1750930">of</span> <span m="1751050">i</span> <span m="1752010">equals</span> <span m="1753180">n</span> <span m="1753460">times</span> <span m="1753760">n</span> <span m="1753880">plus</span> <span m="1754130">1</span> <span m="1754300">over</span> <span m="1754470">2.</span> </p>
<p><span m="1757910">And</span> <span m="1758000">once</span> <span m="1758180">you've</span> <span m="1758290">got</span> <span m="1758450">that</span> <span m="1758630">established,</span> <span m="1759210">now</span> <span m="1759390">we're</span> <span m="1759480">going</span> <span m="1759580">to</span> <span m="1759630">go</span> <span m="1760870">verify</span> <span m="1762660">that</span> <span m="1763330">p</span> <span m="1763490">0</span> <span m="1763830">is</span> <span m="1763940">true</span> <span m="1764300">and that</span> <span m="1764370">p n</span> <span m="1765756">implies</span> <span m="1766218">p n</span> <span m="1766680">plus</span> <span m="1766890">1.</span> <span m="1769030">So</span> <span m="1769210">we</span> <span m="1769490">always</span> <span m="1769760">have</span> <span m="1769860">to</span> <span m="1769950">write</span> <span m="1770130">this</span> <span m="1770320">down.</span> <span m="1771510">The</span> <span m="1771580">next</span> <span m="1771810">thing</span> <span m="1771930">to</span> <span m="1772010">do</span> <span m="1772360">is</span> <span m="1772540">to</span> <span m="1772650">check</span> <span m="1772990">what's</span> <span m="1773170">called</span> <span m="1773500">the</span> <span m="1773560">base</span> <span m="1773930">case,</span> <span m="1774740">p</span> <span m="1775040">zero.</span> <span m="1776165">So</span> <span m="1776600">let's</span> <span m="1776870">do</span> <span m="1777010">that.</span> </p>
<p><span m="1792510">So</span> <span m="1792680">we</span> <span m="1792800">write</span> <span m="1793050">down</span> <span m="1793330">base</span> <span m="1793620">case.</span> <span m="1794130">Some</span> <span m="1794270">people</span> <span m="1794460">call</span> <span m="1794710">it the</span> <span m="1794790">basis</span> <span m="1795280">step.</span> <span m="1799160">And</span> <span m="1799450">we</span> <span m="1799530">have</span> <span m="1799640">to</span> <span m="1799750">check</span> <span m="1800930">that</span> <span m="1801300">p</span> <span m="1801540">0</span> <span m="1802500">is</span> <span m="1802650">true.</span> <span m="1805860">Well,</span> <span m="1807150">what's</span> <span m="1807410">the</span> <span m="1807510">sum</span> <span m="1807930">of</span> <span m="1808050">i</span> <span m="1808550">equals</span> <span m="1809070">1</span> <span m="1809720">to</span> <span m="1809920">0</span> <span m="1810660">of</span> <span m="1810960">i?</span> <span m="1813650">0.</span> <span m="1814650">There</span> <span m="1814820">are</span> <span m="1814920">no</span> <span m="1815190">terms</span> <span m="1815600">in</span> <span m="1815660">this</span> <span m="1815770">sum.</span> <span m="1818750">And</span> <span m="1819030">if</span> <span m="1819140">I</span> <span m="1819230">look</span> <span m="1819470">over</span> <span m="1819650">there,</span> <span m="1820040">n</span> <span m="1820380">times</span> <span m="1820700">n</span> <span m="1820880">plus</span> <span m="1821130">1</span> <span m="1821300">over</span> <span m="1821470">2.</span> <span m="1822020">If n is</span> <span m="1822270">0,</span> <span m="1823982">it</span> <span m="1824430">equals</span> <span m="1824730">0.</span> </p>
<p><span m="1828610">So</span> <span m="1828790">we're</span> <span m="1828880">done</span> <span m="1829090">with</span> <span m="1829180">the</span> <span m="1829250">base</span> <span m="1829510">case.</span> <span m="1829720">We've</span> <span m="1829870">now</span> <span m="1830180">proved</span> <span m="1830670">that</span> <span m="1830780">p0</span> <span m="1831420">is</span> <span m="1831520">true.</span> <span m="1833890">And</span> <span m="1834010">the</span> <span m="1834080">second</span> <span m="1834460">part</span> <span m="1835550">is</span> <span m="1835660">called</span> <span m="1835940">the</span> <span m="1836010">inductive</span> <span m="1836500">step.</span> <span m="1842460">And</span> <span m="1842690">here</span> <span m="1843040">we</span> <span m="1843150">have</span> <span m="1843370">to</span> <span m="1843480">show,</span> <span m="1844510">for</span> <span m="1844750">n</span> <span m="1845150">greater</span> <span m="1845440">than</span> <span m="1845540">or</span> <span m="1845640">equal</span> <span m="1845710">to</span> <span m="1845780">0,</span> <span m="1847350">we</span> <span m="1847430">need</span> <span m="1847610">to</span> <span m="1847670">show</span> <span m="1848160">that</span> <span m="1849360">pn</span> <span m="1850840">implies</span> <span m="1851830">pn</span> <span m="1852190">plus</span> <span m="1852450">1</span> <span m="1852650">is</span> <span m="1852750">true.</span> </p>
<p><span m="1861620">Now</span> <span m="1861790">how</span> <span m="1862050">do</span> <span m="1862100">we</span> <span m="1862220">show</span> <span m="1862620">an</span> <span m="1862690">implication</span> <span m="1863370">is</span> <span m="1863470">true?</span> <span m="1865380">How</span> <span m="1865560">do</span> <span m="1865680">I</span> <span m="1865750">show</span> <span m="1866070">this</span> <span m="1866270">is</span> <span m="1866380">true?</span> <span m="1866580">What</span> <span m="1866710">am</span> <span m="1866810">I</span> <span m="1866870">going</span> <span m="1866990">to</span> <span m="1867080">do</span> <span m="1867320">to</span> <span m="1867390">show</span> <span m="1867660">that's</span> <span m="1867890">true</span> <span m="1868260">in</span> <span m="1868390">general?</span> <span m="1869810">Yeah?</span> </p>
<p><span m="1871290">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="1879060">PROFESSOR: Right.</span> <span m="1879780">Because</span> <span m="1880250">an</span> <span m="1880330">implication</span> <span m="1881010">is</span> <span m="1881120">true</span> <span m="1881770">in</span> <span m="1881970">every</span> <span m="1882300">case,</span> <span m="1882740">except</span> <span m="1883270">for</span> <span m="1884030">true</span> <span m="1884230">implies</span> <span m="1884620">false.</span> </p>
<p><span m="1886060">So</span> <span m="1886260">if</span> <span m="1886430">pn</span> <span m="1886750">is</span> <span m="1886890">false,</span> <span m="1887330">we're</span> <span m="1887490">done.</span> <span m="1887960">This</span> <span m="1888140">implication</span> <span m="1888630">is</span> <span m="1888700">true.</span> <span m="1889410">The</span> <span m="1889550">only</span> <span m="1889730">case</span> <span m="1889980">we</span> <span m="1890050">have</span> <span m="1890140">to</span> <span m="1890240">worry</span> <span m="1890440">about</span> <span m="1890660">is</span> <span m="1890780">p</span> <span m="1890980">of</span> <span m="1891060">n</span> <span m="1891210">is</span> <span m="1891300">true.</span> <span m="1892530">So</span> <span m="1892600">we</span> <span m="1892740">assume</span> <span m="1893220">p</span> <span m="1893410">of</span> <span m="1893480">n</span> <span m="1893620">is</span> <span m="1893720">true.</span> <span m="1894600">And</span> <span m="1894830">now</span> <span m="1895050">we</span> <span m="1895170">confirm</span> <span m="1895930">that</span> <span m="1896060">that</span> <span m="1896250">means</span> <span m="1896500">pn</span> <span m="1896770">plus</span> <span m="1896990">1</span> <span m="1897150">is</span> <span m="1897240">true.</span> <span m="1900920">So</span> <span m="1902120">we</span> <span m="1902230">write</span> <span m="1902450">that</span> <span m="1902650">down.</span> <span m="1903960">Assume</span> <span m="1906080">pn</span> <span m="1907560">is</span> <span m="1907750">true.</span> <span m="1909960">And</span> <span m="1910160">you</span> <span m="1910230">might</span> <span m="1910410">also</span> <span m="1910690">write</span> <span m="1910920">down,</span> <span m="1911690">"for</span> <span m="1912090">purposes</span> <span m="1912680">of</span> <span m="1912800">induction"</span> <span m="1913765">or</span> <span m="1914980">"purposes</span> <span m="1915520">of</span> <span m="1915610">it</span> <span m="1915650">verifying</span> <span m="1915945">the</span> <span m="1916240">inductive</span> <span m="1916650">hypothesis."</span> <span m="1919310">just</span> <span m="1919490">to</span> <span m="1919570">let</span> <span m="1919740">us</span> <span m="1919900">know</span> <span m="1920220">why</span> <span m="1920570">you're</span> <span m="1920710">assuming</span> <span m="1921180">it.</span> </p>
<p><span m="1921330">In</span> <span m="1921410">other</span> <span m="1921550">words,</span> <span m="1921810">you're</span> <span m="1921860">not</span> <span m="1922100">assuming</span> <span m="1922610">pn</span> <span m="1922800">is</span> <span m="1922980">true</span> <span m="1923190">for</span> <span m="1923330">purposes</span> <span m="1923820">of</span> <span m="1923900">contradiction.</span> <span m="1925760">You're</span> <span m="1925880">assuming</span> <span m="1926220">it</span> <span m="1926300">for</span> <span m="1926400">purposes</span> <span m="1927000">of</span> <span m="1927160">induction.</span> <span m="1930370">All</span> <span m="1930500">right,</span> <span m="1930630">let's</span> <span m="1930830">do</span> <span m="1930960">that.</span> <span m="1943150">That</span> <span m="1943420">means,</span> <span m="1943970">in</span> <span m="1944050">this</span> <span m="1944270">case,</span> <span m="1946380">i.e.</span> <span m="1947400">we</span> <span m="1947550">assume</span> <span m="1951270">1</span> <span m="1951590">plus</span> <span m="1951930">2</span> <span m="1952410">plus</span> <span m="1952790">up to</span> <span m="1952880">n</span> <span m="1954370">equals</span> <span m="1956020">n</span> <span m="1956350">times</span> <span m="1956660">n</span> <span m="1956820">plus</span> <span m="1957140">1</span> <span m="1957470">over</span> <span m="1957740">2.</span> </p>
<p><span m="1960270">And</span> <span m="1960420">we</span> <span m="1960510">need</span> <span m="1960740">to</span> <span m="1960800">show</span> <span m="1967436">that</span> <span m="1967930">the</span> <span m="1968190">n</span> <span m="1968310">plus</span> <span m="1968570">1</span> <span m="1968810">case</span> <span m="1969100">is</span> <span m="1969190">true,</span> <span m="1969910">and</span> <span m="1970090">the</span> <span m="1970210">1</span> <span m="1970675">plus 2</span> <span m="1971140">plus</span> <span m="1972740">n</span> <span m="1972940">plus</span> <span m="1973200">1</span> <span m="1974740">equals</span> <span m="1976440">n</span> <span m="1976720">plus</span> <span m="1977120">1</span> <span m="1977550">times</span> <span m="1978130">n</span> <span m="1978350">plus</span> <span m="1978750">2</span> <span m="1980460">over</span> <span m="1980680">2.</span> <span m="1984710">It's</span> <span m="1985070">sort</span> <span m="1985310">of</span> <span m="1985390">weird,</span> <span m="1985980">and</span> <span m="1986400">this</span> <span m="1986660">does</span> <span m="1986930">confuse</span> <span m="1987390">people</span> <span m="1988150">that</span> <span m="1988290">aren't</span> <span m="1988530">familiar</span> <span m="1989070">yet</span> <span m="1989440">with</span> <span m="1989530">induction.</span> <span m="1990830">It</span> <span m="1990950">looks</span> <span m="1991160">like</span> <span m="1991290">we</span> <span m="1991400">just</span> <span m="1991680">assumed</span> <span m="1992090">what we're</span> <span m="1992260">trying</span> <span m="1992540">to</span> <span m="1992610">prove.</span> <span m="1994360">We're</span> <span m="1994460">trying</span> <span m="1994730">to</span> <span m="1994800">prove</span> <span m="1995090">this</span> <span m="1995250">is</span> <span m="1995340">true</span> <span m="1995520">for</span> <span m="1995690">all</span> <span m="1995880">n.</span> <span m="1996410">And</span> <span m="1996600">we</span> <span m="1996690">just</span> <span m="1996850">assumed</span> <span m="1997250">it.</span> </p>
<p><span m="1998280">But</span> <span m="1998450">we're</span> <span m="1998590">assuming</span> <span m="1999090">it in</span> <span m="1999200">the</span> <span m="1999290">context</span> <span m="2000030">of</span> <span m="2000110">establishing</span> <span m="2002360">this</span> <span m="2002620">implication</span> <span m="2003270">is</span> <span m="2003370">true.</span> <span m="2004080">And</span> <span m="2004200">then</span> <span m="2004390">we</span> <span m="2004500">apply</span> <span m="2004820">the</span> <span m="2004950">induction</span> <span m="2005350">axiom</span> <span m="2005980">to</span> <span m="2006090">conclude</span> <span m="2006530">pn</span> <span m="2006880">is true</span> <span m="2007000">for all n.</span> <span m="2010540">All</span> <span m="2010670">right,</span> <span m="2013590">well,</span> <span m="2014070">let's</span> <span m="2014320">rewrite</span> <span m="2014810">this</span> <span m="2016200">as</span> <span m="2017280">1</span> <span m="2017703">plus 2</span> <span m="2019040">plus</span> <span m="2020840">n</span> <span m="2021280">plus</span> <span m="2021760">n</span> <span m="2021960">plus</span> <span m="2022240">1.</span> <span m="2026140">Because</span> <span m="2026570">I've</span> <span m="2026700">assumed</span> <span m="2027200">pn,</span> <span m="2027780">I</span> <span m="2027880">can</span> <span m="2028050">rewrite</span> <span m="2028530">this</span> <span m="2030130">as</span> <span m="2030420">an</span> <span m="2031430">n</span> <span m="2031550">plus</span> <span m="2031820">1</span> <span m="2032140">over</span> <span m="2032460">2.</span> <span m="2034960">And</span> <span m="2035120">now</span> <span m="2036410">plus</span> <span m="2037070">n</span> <span m="2037220">plus</span> <span m="2037480">1</span> <span m="2037780">out</span> <span m="2037910">here.</span> </p>
<p><span m="2039600">That</span> <span m="2039940">equals,</span> <span m="2042490">well,</span> <span m="2042740">I</span> <span m="2042820">got</span> <span m="2043390">n</span> <span m="2043600">squared</span> <span m="2044600">plus</span> <span m="2045200">n,</span> <span m="2046040">here,</span> <span m="2046520">over</span> <span m="2046660">2,</span> <span m="2049179">plus</span> <span m="2049510">2</span> <span m="2049719">n</span> <span m="2049840">plus</span> <span m="2050070">2.</span> <span m="2055290">And</span> <span m="2055420">that</span> <span m="2055650">equals</span> <span m="2057780">n</span> <span m="2057909">plus</span> <span m="2058179">one</span> <span m="2059210">times</span> <span m="2059460">n</span> <span m="2059580">plus</span> <span m="2059830">2</span> <span m="2060719">over</span> <span m="2060940">2,</span> <span m="2062050">which</span> <span m="2062250">is</span> <span m="2062350">what</span> <span m="2062560">we're</span> <span m="2062670">trying</span> <span m="2062940">to</span> <span m="2062989">show.</span> <span m="2065830">So</span> <span m="2065969">we've</span> <span m="2066090">completed,</span> <span m="2067159">now,</span> <span m="2067429">the</span> <span m="2067550">inductive</span> <span m="2067980">step.</span> <span m="2068350">We</span> <span m="2068449">have</span> <span m="2068570">shown</span> <span m="2069159">that</span> <span m="2070429">for</span> <span m="2070630">all</span> <span m="2070870">n,</span> <span m="2071736">pn</span> <span m="2072170">implies</span> <span m="2072909">pn</span> <span m="2073150">plus</span> <span m="2073380">one</span> <span m="2073870">for all n</span> <span m="2074300">greater than or equal to</span> <span m="2074780">0.</span> </p>
<p><span m="2081710">Any</span> <span m="2081920">questions</span> <span m="2085199">about</span> <span m="2085460">that?</span> <span m="2085760">So,</span> <span m="2086580">the</span> <span m="2086679">proof</span> <span m="2086889">is</span> <span m="2087010">done.</span> <span m="2089370">We've</span> <span m="2090250">done</span> <span m="2090429">everything</span> <span m="2090739">we</span> <span m="2090840">need</span> <span m="2091010">to</span> <span m="2091100">imply</span> <span m="2091409">induction.</span> <span m="2091830">We've</span> <span m="2091960">got</span> <span m="2092580">p0</span> <span m="2093040">is</span> <span m="2093130">true.</span> <span m="2093909">And</span> <span m="2094120">pn</span> <span m="2094409">implies</span> <span m="2094760">pn</span> <span m="2095000">plus</span> <span m="2095210">1</span> <span m="2095469">for</span> <span m="2095639">n</span> <span m="2095810">bigger than or</span> <span m="2096230">equal</span> <span m="2096429">to</span> <span m="2096480">zero.</span> </p>
<p><span m="2099650">Now</span> <span m="2101710">induction</span> <span m="2102430">helped</span> <span m="2102790">us</span> <span m="2102910">prove</span> <span m="2103130">the</span> <span m="2103210">theorem.</span> <span m="2105310">Did</span> <span m="2105430">it</span> <span m="2105520">help</span> <span m="2105750">us</span> <span m="2105880">understand</span> <span m="2106330">why</span> <span m="2106430">the</span> <span m="2106540">theorem</span> <span m="2106760">is true?</span> <span m="2109990">Do</span> <span m="2110070">you</span> <span m="2110140">have</span> <span m="2110310">any</span> <span m="2110490">feel</span> <span m="2110890">for</span> <span m="2111000">why</span> <span m="2111220">the</span> <span m="2111320">theorem</span> <span m="2111570">is</span> <span m="2111670">true</span> <span m="2111900">after</span> <span m="2112100">seeing</span> <span m="2112320">the</span> <span m="2112400">proof?</span> <span m="2115410">Not</span> <span m="2115650">really.</span> <span m="2117670">I</span> <span m="2117840">don't</span> <span m="2117980">think--</span> <span m="2118150">sometimes</span> <span m="2119570">induction</span> <span m="2120000">will</span> <span m="2120240">give you</span> <span m="2120410">an</span> <span m="2120490">understanding.</span> <span m="2121000">Sometimes</span> <span m="2121510">it</span> <span m="2121580">won't.</span> <span m="2121900">Here</span> <span m="2122220">you've</span> <span m="2122320">got</span> <span m="2123250">no</span> <span m="2123440">understanding</span> <span m="2123980">of</span> <span m="2124060">why</span> <span m="2124300">the</span> <span m="2124410">theorem's</span> <span m="2124660">true,</span> <span m="2124950">which</span> <span m="2125110">is</span> <span m="2125180">sort</span> <span m="2125370">of</span> <span m="2125450">unfortunate.</span> </p>
<p><span m="2127160">Did</span> <span m="2127330">induction</span> <span m="2127840">help</span> <span m="2128130">you</span> <span m="2128260">figure</span> <span m="2128710">out</span> <span m="2130880">the</span> <span m="2131110">answer</span> <span m="2131530">to</span> <span m="2131630">the</span> <span m="2131740">sum?</span> <span m="2133100">Namely,</span> <span m="2133360">say</span> <span m="2133570">you</span> <span m="2133650">were</span> <span m="2133730">trying</span> <span m="2133980">to</span> <span m="2134050">derive</span> <span m="2134520">this</span> <span m="2134740">answer</span> <span m="2135190">from</span> <span m="2135380">this</span> <span m="2135500">sum.</span> <span m="2135810">Did</span> <span m="2135920">induction</span> <span m="2136490">give</span> <span m="2136630">you</span> <span m="2136730">the</span> <span m="2136860">answer?</span> <span m="2139730">No.</span> <span m="2140450">You</span> <span m="2140620">had</span> <span m="2140810">to</span> <span m="2140890">know</span> <span m="2141450">the</span> <span m="2141620">answer,</span> <span m="2142510">namely</span> <span m="2143410">this,</span> <span m="2144430">in</span> <span m="2144580">order</span> <span m="2144720">to</span> <span m="2144800">prove</span> <span m="2145060">it</span> <span m="2145110">was</span> <span m="2145260">true.</span> <span m="2147320">Now</span> <span m="2147470">later</span> <span m="2147810">we'll</span> <span m="2147920">see</span> <span m="2148110">examples</span> <span m="2148590">where</span> <span m="2149170">induction</span> <span m="2149580">actually</span> <span m="2149830">can</span> <span m="2149990">give</span> <span m="2150150">you</span> <span m="2150240">the</span> <span m="2150400">answer,</span> <span m="2150690">but</span> <span m="2150840">often</span> <span m="2151220">it</span> <span m="2151300">does</span> <span m="2151480">not.</span> </p>
<p><span m="2152380">Often,</span> <span m="2152760">induction</span> <span m="2153330">gives</span> <span m="2153480">you</span> <span m="2153560">no</span> <span m="2153790">hints,</span> <span m="2154660">no</span> <span m="2154880">answer,</span> <span m="2155280">just</span> <span m="2155620">prove</span> <span m="2155950">that it's</span> <span m="2156170">right</span> <span m="2156710">once</span> <span m="2157070">you</span> <span m="2157160">had</span> <span m="2157370">the</span> <span m="2157440">clever</span> <span m="2157920">idea</span> <span m="2158780">that,</span> <span m="2159090">oh,</span> <span m="2159230">maybe</span> <span m="2159450">that's</span> <span m="2159660">the</span> <span m="2159790">answer.</span> <span m="2161940">You'll</span> <span m="2162050">see</span> <span m="2162210">that</span> <span m="2162370">with</span> <span m="2162490">things</span> <span m="2162690">like</span> <span m="2162820">the</span> <span m="2162890">beaver</span> <span m="2163210">flu</span> <span m="2163440">problem.</span> <span m="2166080">Figuring</span> <span m="2166840">out</span> <span m="2167140">the</span> <span m="2167240">inductive</span> <span m="2167590">hypothesis</span> <span m="2168410">or</span> <span m="2169000">the</span> <span m="2169110">answer,</span> <span m="2170040">you</span> <span m="2170150">know,</span> <span m="2170300">the</span> <span m="2170430">details</span> <span m="2170900">of</span> <span m="2171020">it</span> <span m="2171300">is</span> <span m="2171540">hard.</span> <span m="2172040">But</span> <span m="2172050">once</span> <span m="2172270">you</span> <span m="2172360">do</span> <span m="2172550">it,</span> <span m="2172650">then</span> <span m="2172890">applying the</span> <span m="2173330">induction,</span> <span m="2174840">not</span> <span m="2175030">so</span> <span m="2175160">hard.</span> <span m="2175530">It</span> <span m="2175620">gives</span> <span m="2175770">you</span> <span m="2175860">a</span> <span m="2175920">concrete</span> <span m="2176320">proof.</span> </p>
<p><span m="2179450">OK,</span> <span m="2179660">let's</span> <span m="2179820">do</span> <span m="2179920">another</span> <span m="2180190">one.</span> <span m="2186070">In</span> <span m="2186260">fact,</span> <span m="2186490">we're</span> <span m="2186600">just</span> <span m="2186790">going</span> <span m="2186890">to</span> <span m="2186950">spend</span> <span m="2187260">the</span> <span m="2187310">rest</span> <span m="2187500">of</span> <span m="2187560">today</span> <span m="2187850">doing</span> <span m="2188270">induction</span> <span m="2188660">proofs.</span> <span m="2195380">So</span> <span m="2199410">for</span> <span m="2199660">all</span> <span m="2199920">natural</span> <span m="2200340">numbers</span> <span m="2200760">n,</span> <span m="2204300">3</span> <span m="2204970">divides</span> <span m="2205830">n</span> <span m="2206080">cubed</span> <span m="2207080">minus</span> <span m="2207430">n.</span> <span m="2208780">Means</span> <span m="2209140">than</span> <span m="2209320">n cubed</span> <span m="2209410">minus</span> <span m="2209670">n is</span> <span m="2209850">a</span> <span m="2209890">multiple</span> <span m="2210210">of</span> <span m="2210290">three.</span> <span m="2211740">For</span> <span m="2211890">example,</span> <span m="2213180">n</span> <span m="2213370">is</span> <span m="2213500">5.</span> <span m="2218220">3</span> <span m="2218810">divides</span> <span m="2219900">125</span> <span m="2221360">minus</span> <span m="2221660">5,</span> <span m="2223450">because</span> <span m="2223590">that's</span> <span m="2223690">120.</span> </p>
<p><span m="2226720">Let's</span> <span m="2226960">prove</span> <span m="2227280">that.</span> <span m="2229340">And</span> <span m="2229660">we're</span> <span m="2229750">going</span> <span m="2229850">to</span> <span m="2229900">use</span> <span m="2230120">induction.</span> <span m="2237620">What</span> <span m="2237770">do</span> <span m="2237810">you</span> <span m="2237910">suppose</span> <span m="2238280">pn's</span> <span m="2238730">going</span> <span m="2238850">to</span> <span m="2238900">be?</span> <span m="2242000">What's</span> <span m="2242160">my</span> <span m="2242300">predicate</span> <span m="2242770">or</span> <span m="2242830">my</span> <span m="2243620">inductive</span> <span m="2244010">hypothesis</span> <span m="2244740">here?</span> <span m="2246810">Any</span> <span m="2246980">thoughts?</span> <span m="2250390">Yeah?</span> </p>
<p><span m="2251752">PROFESSOR: [INAUDIBLE]</span> </p>
<p><span m="2253460">PROFESSOR: Yeah, that's you.</span> <span m="2253850">Go</span> <span m="2253950">ahead.</span> </p>
<p><span m="2254638">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="2256270">PROFESSOR: Yeah.</span> <span m="2256890">First</span> <span m="2257310">thing</span> <span m="2257490">you</span> <span m="2257690">always</span> <span m="2258110">want</span> <span m="2258270">to</span> <span m="2258320">try</span> <span m="2258710">is</span> <span m="2259050">you</span> <span m="2259210">assume</span> <span m="2259590">that</span> <span m="2259780">is your</span> <span m="2259970">induction</span> <span m="2260340">hypothesis.</span> <span m="2262460">So</span> <span m="2263250">we</span> <span m="2263380">say,</span> <span m="2264020">let</span> <span m="2264220">pn</span> <span m="2266840">be</span> <span m="2267470">3</span> <span m="2268140">divides</span> <span m="2269390">n</span> <span m="2269580">cubed</span> <span m="2269870">minus</span> <span m="2270180">n.</span> <span m="2273040">What's</span> <span m="2273360">the</span> <span m="2273440">next</span> <span m="2273660">thing</span> <span m="2273790">we</span> <span m="2273880">do</span> <span m="2274020">in</span> <span m="2274090">our</span> <span m="2274180">proof?</span> <span m="2276230">Base</span> <span m="2276570">case.</span> <span m="2279540">Always</span> <span m="2279850">easy</span> <span m="2280100">to</span> <span m="2280180">forget,</span> <span m="2280630">but</span> <span m="2280850">not</span> <span m="2281050">a</span> <span m="2281290">good</span> <span m="2281490">idea.</span> <span m="2283200">Base</span> <span m="2283450">case</span> <span m="2283720">is</span> <span m="2283820">n</span> <span m="2283950">equals</span> <span m="2284250">0.</span> <span m="2285550">And</span> <span m="2285770">sure</span> <span m="2286060">enough,</span> <span m="2286470">3</span> <span m="2286720">divides</span> <span m="2287900">0</span> <span m="2288240">minus</span> <span m="2288590">0.</span> <span m="2289850">So</span> <span m="2289990">that's</span> <span m="2290260">done.</span> </p>
<p><span m="2298650">What's</span> <span m="2298910">the</span> <span m="2299050">next</span> <span m="2299450">step</span> <span m="2299760">we</span> <span m="2299870">do?</span> <span m="2304170">What</span> <span m="2304320">is</span> <span m="2304420">it?</span> <span m="2304590">Next</span> <span m="2304810">step?</span> <span m="2308710">Inductive</span> <span m="2309320">step.</span> <span m="2316940">And</span> <span m="2317310">for</span> <span m="2317400">that</span> <span m="2318700">we're</span> <span m="2318850">going</span> <span m="2318980">to</span> <span m="2319690">need</span> <span m="2319850">to</span> <span m="2319910">show</span> <span m="2320310">for</span> <span m="2320530">n</span> <span m="2321750">bigger</span> <span m="2321960">than or</span> <span m="2322070">equal</span> <span m="2322280">to</span> <span m="2322340">0,</span> <span m="2323850">we</span> <span m="2323920">want</span> <span m="2324060">to</span> <span m="2324110">show</span> <span m="2325410">pn</span> <span m="2326930">implies</span> <span m="2327650">pn</span> <span m="2328120">plus</span> <span m="2328400">1</span> <span m="2329760">is</span> <span m="2329910">true.</span> </p>
<p><span m="2334880">So</span> <span m="2335040">to</span> <span m="2335150">do</span> <span m="2335340">that</span> <span m="2335610">we</span> <span m="2335760">assume</span> <span m="2336320">pn is</span> <span m="2336790">true.</span> <span m="2344750">In</span> <span m="2344880">other</span> <span m="2345060">words,</span> <span m="2345510">we</span> <span m="2345560">assume</span> <span m="2347570">that</span> <span m="2347680">3</span> <span m="2348160">divides</span> <span m="2349170">n</span> <span m="2349400">cubed</span> <span m="2349730">minus</span> <span m="2350100">n.</span> <span m="2351590">And</span> <span m="2351730">we're</span> <span m="2351820">trying</span> <span m="2352150">to</span> <span m="2352200">show</span> <span m="2352700">that</span> <span m="2353290">3</span> <span m="2353620">divides</span> <span m="2354110">n</span> <span m="2354310">plus</span> <span m="2354610">1</span> <span m="2354980">cubed</span> <span m="2355380">minus</span> <span m="2355690">n</span> <span m="2355820">plus</span> <span m="2356090">1.</span> <span m="2357270">So</span> <span m="2357510">we</span> <span m="2358130">take</span> <span m="2358310">a</span> <span m="2358370">look</span> <span m="2358580">at,</span> <span m="2358750">we</span> <span m="2358850">examine</span> <span m="2362130">n</span> <span m="2362360">plus</span> <span m="2362670">1</span> <span m="2362950">cubed</span> <span m="2363870">minus</span> <span m="2364290">n</span> <span m="2364430">plus</span> <span m="2364680">1.</span> <span m="2365050">And we</span> <span m="2365160">want</span> <span m="2365370">to</span> <span m="2365420">show</span> <span m="2365700">it's</span> <span m="2365810">a</span> <span m="2365850">multiple</span> <span m="2366210">of</span> <span m="2366300">three.</span> </p>
<p><span m="2369070">Well,</span> <span m="2369520">let's</span> <span m="2369900">expand</span> <span m="2370320">that</span> <span m="2370490">out.</span> <span m="2372540">This</span> <span m="2372840">is</span> <span m="2373870">n</span> <span m="2374140">cubed</span> <span m="2374610">plus</span> <span m="2374950">3n</span> <span m="2375420">squared,</span> <span m="2376110">plus</span> <span m="2376420">3n</span> <span m="2377420">plus</span> <span m="2377740">1.</span> <span m="2379160">And</span> <span m="2379300">I</span> <span m="2379350">subtract</span> <span m="2379960">off</span> <span m="2381380">n</span> <span m="2381510">plus</span> <span m="2381770">1.</span> <span m="2384100">So</span> <span m="2384290">I</span> <span m="2384380">get</span> <span m="2385200">n</span> <span m="2385440">cubed</span> <span m="2385970">plus</span> <span m="2386350">3n</span> <span m="2386780">squared</span> <span m="2388060">plus</span> <span m="2388460">2n.</span> <span m="2394830">Is</span> <span m="2395020">this</span> <span m="2395200">a</span> <span m="2395260">multiple</span> <span m="2395570">of</span> <span m="2395660">3?</span> <span m="2399600">I</span> <span m="2399680">need</span> <span m="2399840">to</span> <span m="2399890">show</span> <span m="2400060">that's</span> <span m="2400240">a</span> <span m="2400280">multiple</span> <span m="2400590">of</span> <span m="2400670">3</span> <span m="2401000">and</span> <span m="2401170">then I'd</span> <span m="2401320">be</span> <span m="2401430">all</span> <span m="2401590">done.</span> <span m="2404220">Is</span> <span m="2404350">it</span> <span m="2404420">a</span> <span m="2404470">multiple</span> <span m="2404740">of</span> <span m="2404820">3?</span> </p>
<p><span m="2408000">Beats</span> <span m="2408240">me.</span> <span m="2409950">It</span> <span m="2410010">doesn't</span> <span m="2410240">look</span> <span m="2410520">like</span> <span m="2410680">a</span> <span m="2410730">multiple</span> <span m="2411080">of</span> <span m="2411170">3,</span> <span m="2412580">necessarily.</span> <span m="2416350">So</span> <span m="2417660">maybe</span> <span m="2417860">we</span> <span m="2417970">need</span> <span m="2418110">to</span> <span m="2418170">massage</span> <span m="2418445">it</span> <span m="2418720">a</span> <span m="2418790">little</span> <span m="2418980">bit.</span> <span m="2419200">I</span> <span m="2419410">do</span> <span m="2419680">know</span> <span m="2419990">that</span> <span m="2420640">this</span> <span m="2420990">is</span> <span m="2421120">a</span> <span m="2421170">multiple</span> <span m="2421570">of</span> <span m="2421660">3.</span> <span m="2421890">I</span> <span m="2421950">can</span> <span m="2422110">use</span> <span m="2422370">that</span> <span m="2422640">fact.</span> <span m="2423690">And</span> <span m="2423860">in</span> <span m="2423920">proofs</span> <span m="2424150">by</span> <span m="2424320">induction</span> <span m="2424740">you</span> <span m="2424850">always</span> <span m="2425120">want</span> <span m="2425290">to--</span> <span m="2426020">if you're</span> <span m="2426110">not</span> <span m="2426310">making</span> <span m="2426530">use of</span> <span m="2426700">that</span> <span m="2426960">fact,</span> <span m="2427290">then</span> <span m="2427370">you're</span> <span m="2427470">not</span> <span m="2427690">really</span> <span m="2427880">making</span> <span m="2428160">use</span> <span m="2428320">of</span> <span m="2428410">induction.</span> <span m="2429390">Sort</span> <span m="2429630">of</span> <span m="2429710">a</span> <span m="2429780">warning</span> <span m="2430100">sign.</span> </p>
<p><span m="2430390">So</span> <span m="2430590">I</span> <span m="2430670">want</span> <span m="2430870">to</span> <span m="2430910">use</span> <span m="2431140">this</span> <span m="2431360">fact</span> <span m="2432030">to</span> <span m="2432190">prove</span> <span m="2432480">this.</span> <span m="2434220">Well,</span> <span m="2435960">let's</span> <span m="2437050">get</span> <span m="2437180">a</span> <span m="2437220">minus</span> <span m="2437485">n</span> <span m="2437810">in</span> <span m="2437940">here.</span> <span m="2439520">This</span> <span m="2439720">equals</span> <span m="2440020">n</span> <span m="2440190">cubed</span> <span m="2440510">minus</span> <span m="2440850">n</span> <span m="2441640">plus</span> <span m="2442240">3n</span> <span m="2442670">squared</span> <span m="2443610">plus--</span> <span m="2444700">if</span> <span m="2444780">I</span> <span m="2444840">subtracted</span> <span m="2445130">an</span> <span m="2445540">n</span> <span m="2445710">I</span> <span m="2445750">got</span> <span m="2445880">to</span> <span m="2445970">add</span> <span m="2446280">an</span> <span m="2446390">n--</span> <span m="2446650">plus</span> <span m="2446910">3n.</span> <span m="2449070">So</span> <span m="2449200">I've</span> <span m="2449340">rewritten</span> <span m="2449790">it.</span> <span m="2450700">Now</span> <span m="2451030">is</span> <span m="2451140">it</span> <span m="2451200">clear?</span> <span m="2453160">Now</span> <span m="2453540">it's</span> <span m="2453680">clear.</span> </p>
<p><span m="2454740">Very</span> <span m="2454980">simple,</span> <span m="2455350">because</span> <span m="2455680">3</span> <span m="2455960">divide</span> <span m="2456320">this</span> <span m="2457250">by</span> <span m="2457440">pn.</span> <span m="2458390">3</span> <span m="2458610">divides</span> <span m="2458960">that.</span> <span m="2459330">And</span> <span m="2459440">3</span> <span m="2459600">divides</span> <span m="2459940">that.</span> <span m="2460810">So</span> <span m="2460960">this</span> <span m="2461420">is</span> <span m="2461590">a</span> <span m="2461650">multiple</span> <span m="2462040">of</span> <span m="2462140">3,</span> <span m="2464420">because</span> <span m="2464960">I've</span> <span m="2465050">got</span> <span m="2465650">3</span> <span m="2466070">divides</span> <span m="2466730">3n</span> <span m="2467140">squared,</span> <span m="2468210">3</span> <span m="2468530">divides</span> <span m="2469200">3n,</span> <span m="2470450">and</span> <span m="2470580">3</span> <span m="2470800">divides</span> <span m="2471920">n</span> <span m="2472150">cubed</span> <span m="2472670">minus</span> <span m="2473070">n</span> <span m="2473990">by</span> <span m="2475730">pn.</span> <span m="2477120">Or</span> <span m="2477290">you</span> <span m="2477360">could</span> <span m="2477490">say,</span> <span m="2477660">by</span> <span m="2477870">the</span> <span m="2477960">inductive</span> <span m="2478320">hypothesis.</span> <span m="2480640">pn</span> <span m="2481130">is</span> <span m="2481420">the</span> <span m="2481540">inductive</span> <span m="2481960">hypothesis,</span> <span m="2483300">another</span> <span m="2483530">name</span> <span m="2483810">for</span> <span m="2484110">it.</span> </p>
<p><span m="2485680">So</span> <span m="2485790">therefore,</span> <span m="2486190">3</span> <span m="2486410">divides</span> <span m="2486830">that,</span> <span m="2487520">which</span> <span m="2487780">means</span> <span m="2488040">3</span> <span m="2488190">divides</span> <span m="2488710">this.</span> <span m="2490820">So</span> <span m="2490940">that</span> <span m="2491120">means</span> <span m="2491360">3</span> <span m="2491550">divides</span> <span m="2492240">n</span> <span m="2492360">plus</span> <span m="2492620">1</span> <span m="2492830">cubed</span> <span m="2494460">minus</span> <span m="2494780">n</span> <span m="2494900">plus</span> <span m="2495160">1.</span> <span m="2496590">And</span> <span m="2496990">we</span> <span m="2497150">are</span> <span m="2497300">done</span> <span m="2499190">with</span> <span m="2499490">the</span> <span m="2499590">proof</span> <span m="2499820">by</span> <span m="2499960">induction.</span> <span m="2502650">Any</span> <span m="2502820">questions</span> <span m="2503280">about that one?</span> </p>
<p><span m="2507680">So</span> <span m="2507830">the</span> <span m="2507950">key</span> <span m="2508200">steps in</span> <span m="2508580">induction</span> <span m="2509110">are</span> <span m="2509210">always</span> <span m="2509490">the</span> <span m="2509590">same.</span> <span m="2510600">You</span> <span m="2510750">write</span> <span m="2510990">down</span> <span m="2511700">"proof</span> <span m="2511990">by</span> <span m="2512170">induction."</span> <span m="2513490">You</span> <span m="2513640">identify</span> <span m="2514260">your</span> <span m="2514410">predicate.</span> <span m="2515570">You</span> <span m="2515700">do</span> <span m="2515860">the</span> <span m="2515980">base</span> <span m="2516310">case,</span> <span m="2517650">usually</span> <span m="2518000">n</span> <span m="2518130">equals</span> <span m="2518380">0,</span> <span m="2518610">but</span> <span m="2518720">it</span> <span m="2518790">could</span> <span m="2518900">be</span> <span m="2518990">something</span> <span m="2519300">else.</span> <span m="2520140">And</span> <span m="2520270">then</span> <span m="2520390">you</span> <span m="2520480">do</span> <span m="2520570">your</span> <span m="2520720">inductive</span> <span m="2521100">step.</span> </p>
<p><span m="2523850">Now</span> <span m="2524030">in</span> <span m="2524130">general,</span> <span m="2524700">you</span> <span m="2524840">could</span> <span m="2524980">start</span> <span m="2525330">your</span> <span m="2525450">induction--</span> <span m="2526480">you don't have to start it</span> <span m="2526860">at</span> <span m="2527270">0,</span> <span m="2527560">you could</span> <span m="2527770">start it</span> <span m="2528240">at</span> <span m="2528570">some</span> <span m="2528860">value</span> <span m="2529300">b,</span> <span m="2530290">some</span> <span m="2530510">integer</span> <span m="2530923">b.</span> <span m="2536610">Let's</span> <span m="2537040">take</span> <span m="2537220">a</span> <span m="2537280">look</span> <span m="2537440">at</span> <span m="2537500">that.</span> </p>
<p><span m="2545050">So</span> <span m="2545430">you</span> <span m="2545590">could</span> <span m="2545860">have</span> <span m="2546070">for</span> <span m="2546140">the</span> <span m="2546230">base</span> <span m="2546550">case,</span> <span m="2549900">you</span> <span m="2550040">could</span> <span m="2550150">have</span> <span m="2550530">p</span> <span m="2550870">of</span> <span m="2550950">b</span> <span m="2551820">is</span> <span m="2551980">true,</span> <span m="2552350">not</span> <span m="2552580">p</span> <span m="2552730">of</span> <span m="2552790">0.</span> <span m="2555130">And</span> <span m="2555310">then</span> <span m="2555510">for</span> <span m="2555610">your</span> <span m="2555800">induction</span> <span m="2556290">step</span> <span m="2560550">you</span> <span m="2560730">would</span> <span m="2560860">have</span> <span m="2561770">for</span> <span m="2562110">all</span> <span m="2562500">n</span> <span m="2562880">bigger</span> <span m="2563200">than</span> <span m="2563370">or equal</span> <span m="2563590">to</span> <span m="2563660">b,</span> <span m="2566460">pn</span> <span m="2566720">implies</span> <span m="2567440">pn</span> <span m="2567750">plus</span> <span m="2567980">1.</span> <span m="2570000">And</span> <span m="2570200">then</span> <span m="2570340">your</span> <span m="2570490">conclusion</span> <span m="2574960">is</span> <span m="2575430">that</span> <span m="2575710">for</span> <span m="2575880">all</span> <span m="2576270">n</span> <span m="2576740">bigger</span> <span m="2577000">than or</span> <span m="2577080">equal</span> <span m="2577270">to</span> <span m="2577340">b,</span> <span m="2578290">pn is</span> <span m="2578710">true.</span> </p>
<p><span m="2580790">So</span> <span m="2580960">inductions</span> <span m="2581480">don't</span> <span m="2581730">always</span> <span m="2582220">have</span> <span m="2582350">to</span> <span m="2582440">start</span> <span m="2582670">at</span> <span m="2582760">0.</span> <span m="2583630">You</span> <span m="2583810">can</span> <span m="2583980">pick</span> <span m="2584190">where</span> <span m="2584300">you</span> <span m="2584410">start.</span> <span m="2585790">Just</span> <span m="2586130">make</span> <span m="2586310">sure</span> <span m="2586670">that</span> <span m="2588130">you</span> <span m="2588280">verify</span> <span m="2588840">the</span> <span m="2588940">starting</span> <span m="2589280">point</span> <span m="2590150">and</span> <span m="2590280">you</span> <span m="2590390">verify</span> <span m="2590880">the</span> <span m="2591000">implication</span> <span m="2591850">for</span> <span m="2591990">all</span> <span m="2592230">n</span> <span m="2592940">at</span> <span m="2593110">that</span> <span m="2593290">starting</span> <span m="2593620">point</span> <span m="2594060">and</span> <span m="2594280">beyond.</span> </p>
<p><span m="2600680">All</span> <span m="2600860">right,</span> <span m="2601020">let's</span> <span m="2601330">now</span> <span m="2603240">do</span> <span m="2603430">a</span> <span m="2603540">false</span> <span m="2604050">proof</span> <span m="2605180">using</span> <span m="2605560">induction.</span> <span m="2615090">We're</span> <span m="2615270">going</span> <span m="2615380">to</span> <span m="2615460">prove</span> <span m="2620160">that</span> <span m="2620550">all</span> <span m="2620830">horses</span> <span m="2621330">are</span> <span m="2621370">the</span> <span m="2621490">same</span> <span m="2621760">color.</span> <span m="2634600">So</span> <span m="2634750">let's</span> <span m="2634980">go</span> <span m="2635120">through</span> <span m="2635230">the</span> <span m="2635330">process.</span> <span m="2637270">So</span> <span m="2637430">the</span> <span m="2637510">proof,</span> <span m="2638010">we</span> <span m="2638120">write</span> <span m="2638350">down</span> <span m="2638580">"by</span> <span m="2638780">induction."</span> </p>
<p><span m="2644750">Now</span> <span m="2644930">we</span> <span m="2645020">need</span> <span m="2645200">our</span> <span m="2645320">induction</span> <span m="2645660">hypothesis,</span> <span m="2646420">the</span> <span m="2646520">predicate.</span> <span m="2648320">So</span> <span m="2649190">we're</span> <span m="2649330">going</span> <span m="2649430">to</span> <span m="2649490">let</span> <span m="2649680">that</span> <span m="2649930">be--</span> <span m="2651670">we</span> <span m="2651770">can't</span> <span m="2651990">let</span> <span m="2652130">it</span> <span m="2652190">be</span> <span m="2652330">this.</span> <span m="2652550">Why</span> <span m="2652750">can't</span> <span m="2653050">this</span> <span m="2653250">be</span> <span m="2653400">our</span> <span m="2653480">predicate?</span> <span m="2654950">All</span> <span m="2655190">horses</span> <span m="2655580">are</span> <span m="2655630">the</span> <span m="2655730">same</span> <span m="2655970">color.</span> <span m="2656340">What's</span> <span m="2656570">wrong</span> <span m="2656890">with</span> <span m="2657000">making</span> <span m="2657280">that</span> <span m="2657550">be</span> <span m="2657650">the</span> <span m="2657750">induction</span> <span m="2658080">hypothesis?</span> <span m="2661090">You</span> <span m="2661200">can't</span> <span m="2661440">plug</span> <span m="2661700">anything</span> <span m="2662030">into</span> <span m="2662200">it.</span> <span m="2662250">There's</span> <span m="2662400">no</span> <span m="2662570">number</span> <span m="2663200">here.</span> <span m="2663900">I've</span> <span m="2663990">got</span> <span m="2664210">to</span> <span m="2664250">have</span> <span m="2664410">a</span> <span m="2664480">number,</span> <span m="2664810">n,</span> <span m="2665470">to</span> <span m="2665590">induct</span> <span m="2665960">on.</span> </p>
<p><span m="2667050">So</span> <span m="2667510">what</span> <span m="2667610">I'm</span> <span m="2667690">going</span> <span m="2667770">to</span> <span m="2667840">say</span> <span m="2668190">is</span> <span m="2668350">that</span> <span m="2670010">in</span> <span m="2670320">any</span> <span m="2670610">set</span> <span m="2673112">of</span> <span m="2673540">n</span> <span m="2673740">horses--</span> <span m="2677000">and</span> <span m="2677200">let's</span> <span m="2677390">make</span> <span m="2677660">n</span> <span m="2677930">bigger</span> <span m="2678230">than or</span> <span m="2678330">equal</span> <span m="2678420">to</span> <span m="2678490">1--</span> <span m="2682180">the</span> <span m="2683590">horses</span> <span m="2686310">are</span> <span m="2687460">all</span> <span m="2688160">the</span> <span m="2688250">same</span> <span m="2688500">color.</span> <span m="2695600">All</span> <span m="2695720">right,</span> <span m="2695860">now</span> <span m="2695970">if</span> <span m="2696060">I</span> <span m="2696180">prove</span> <span m="2696650">this</span> <span m="2696830">is</span> <span m="2696940">true</span> <span m="2697160">for</span> <span m="2697370">all</span> <span m="2697610">n,</span> <span m="2698610">well</span> <span m="2698790">then</span> <span m="2698970">all</span> <span m="2699150">horses</span> <span m="2699530">are</span> <span m="2699570">the</span> <span m="2699680">same</span> <span m="2699930">color.</span> <span m="2700280">Because</span> <span m="2700440">in</span> <span m="2700520">any</span> <span m="2700730">set</span> <span m="2700960">of</span> <span m="2701040">n</span> <span m="2701170">horses,</span> <span m="2701550">they're</span> <span m="2701630">all</span> <span m="2701720">the</span> <span m="2701800">same</span> <span m="2702020">color.</span> <span m="2702360">So</span> <span m="2703330">all</span> <span m="2703530">horses</span> <span m="2703870">must</span> <span m="2704110">be</span> <span m="2704170">the</span> <span m="2704270">same</span> <span m="2704530">color.</span> </p>
<p><span m="2705860">What's</span> <span m="2706130">the</span> <span m="2706210">next</span> <span m="2706410">thing</span> <span m="2706570">I</span> <span m="2706620">do?</span> <span m="2707020">What's</span> <span m="2707090">the</span> <span m="2707150">next</span> <span m="2707360">step?</span> <span m="2709150">Base</span> <span m="2709490">case.</span> <span m="2711870">Now,</span> <span m="2712150">what</span> <span m="2712490">am</span> <span m="2712570">I</span> <span m="2712640">going</span> <span m="2712760">to</span> <span m="2712800">use</span> <span m="2713100">as</span> <span m="2713200">my</span> <span m="2713330">base</span> <span m="2713820">case</span> <span m="2714100">here?</span> <span m="2716790">One</span> <span m="2717010">horse,</span> <span m="2718250">OK,</span> <span m="2718530">or</span> <span m="2718660">n</span> <span m="2718840">equals</span> <span m="2719130">1.</span> <span m="2721580">So</span> <span m="2721780">p</span> <span m="2722020">of</span> <span m="2722110">1.</span> <span m="2722920">That</span> <span m="2723100">would</span> <span m="2723280">say</span> <span m="2723590">that</span> <span m="2723820">any</span> <span m="2724160">set</span> <span m="2724570">of</span> <span m="2724770">one</span> <span m="2724980">horses</span> <span m="2726220">the</span> <span m="2726310">horses</span> <span m="2726650">are</span> <span m="2726750">all</span> <span m="2726860">the</span> <span m="2726940">same</span> <span m="2727170">color.</span> </p>
<p><span m="2728510">That's</span> <span m="2728770">true.</span> <span m="2729050">I've</span> <span m="2729140">got one</span> <span m="2729320">horse.</span> <span m="2729650">It's</span> <span m="2729830">the</span> <span m="2729930">same</span> <span m="2730350">color as</span> <span m="2730730">itself.</span> <span m="2732180">So</span> <span m="2732690">that's</span> <span m="2732900">easy.</span> <span m="2733970">It's</span> <span m="2734190">true</span> <span m="2735960">since</span> <span m="2737420">just</span> <span m="2737680">one</span> <span m="2737820">horse.</span> <span m="2763240">All right,</span> <span m="2763370">what's</span> <span m="2763620">the</span> <span m="2763750">next</span> <span m="2764000">step</span> <span m="2766390">of</span> <span m="2766510">the</span> <span m="2766600">proof?</span> <span m="2768100">What's</span> <span m="2768140">the next</span> <span m="2768320">thing</span> <span m="2768450">I</span> <span m="2768510">do?</span> <span m="2770200">Inductive</span> <span m="2770680">step.</span> </p>
<p><span m="2775440">So</span> <span m="2776600">I'm</span> <span m="2776820">going</span> <span m="2777090">to</span> <span m="2777340">assume</span> <span m="2780090">that</span> <span m="2780310">pn</span> <span m="2780680">is</span> <span m="2780800">true</span> <span m="2783760">to</span> <span m="2784130">prove</span> <span m="2787040">pn--</span> <span m="2787510">and</span> <span m="2787800">show</span> <span m="2788060">pn</span> <span m="2788920">plus</span> <span m="2789100">1</span> <span m="2789260">is</span> <span m="2789360">true.</span> <span m="2792030">All</span> <span m="2792230">right,</span> <span m="2793560">so</span> <span m="2793700">I'm</span> <span m="2793810">going</span> <span m="2794030">to</span> <span m="2794230">assume</span> <span m="2794630">that</span> <span m="2794790">in any</span> <span m="2795010">set</span> <span m="2795220">of</span> <span m="2795300">n horses,</span> <span m="2795860">the</span> <span m="2795950">horses</span> <span m="2796310">are</span> <span m="2796410">all</span> <span m="2796500">the</span> <span m="2796570">same</span> <span m="2796810">color</span> <span m="2797680">that</span> <span m="2797810">I</span> <span m="2797880">start</span> <span m="2798180">with.</span> <span m="2798680">And</span> <span m="2798870">now</span> <span m="2799050">I</span> <span m="2799140">look</span> <span m="2799370">at</span> <span m="2799430">a</span> <span m="2799490">set</span> <span m="2799760">of</span> <span m="2800200">n</span> <span m="2800360">plus</span> <span m="2800590">1</span> <span m="2800770">horses.</span> </p>
<p><span m="2804160">So</span> <span m="2804260">we</span> <span m="2804370">consider</span> <span m="2807150">a</span> <span m="2807300">set</span> <span m="2809260">of</span> <span m="2809450">n</span> <span m="2809690">plus</span> <span m="2809980">1</span> <span m="2810170">horses.</span> <span m="2813290">And</span> <span m="2813470">let's</span> <span m="2813680">call</span> <span m="2814040">those</span> <span m="2814270">horses</span> <span m="2814750">h1,</span> <span m="2815940">h2,</span> <span m="2818894">hn</span> <span m="2819380">plus</span> <span m="2819590">1.</span> <span m="2823820">What</span> <span m="2824040">do</span> <span m="2824160">I</span> <span m="2824240">know</span> <span m="2824610">about</span> <span m="2826740">horses</span> <span m="2827350">h1</span> <span m="2827610">to</span> <span m="2828060">hn?</span> <span m="2833740">They're</span> <span m="2833820">the</span> <span m="2833930">same</span> <span m="2834210">color,</span> <span m="2835680">because</span> <span m="2835970">pn</span> <span m="2836320">is</span> <span m="2836440">true.</span> <span m="2837340">There are</span> <span m="2837460">a</span> <span m="2837840">set</span> <span m="2838030">of</span> <span m="2838100">n horses.</span> <span m="2839290">By</span> <span m="2839730">p1</span> <span m="2840260">they're</span> <span m="2840400">all</span> <span m="2840500">the</span> <span m="2840580">same</span> <span m="2840800">color.</span> </p>
<p><span m="2851810">Also,</span> <span m="2854340">what</span> <span m="2854560">do</span> <span m="2854640">I</span> <span m="2854710">know</span> <span m="2854930">but</span> <span m="2855170">h2,</span> <span m="2856570">h3</span> <span m="2858725">and</span> <span m="2859156">hn</span> <span m="2859590">plus</span> <span m="2859860">1?</span> <span m="2862310">All</span> <span m="2862550">the</span> <span m="2862640">same</span> <span m="2862860">color,</span> <span m="2863180">because</span> <span m="2863360">they're</span> <span m="2863490">a</span> <span m="2863550">set</span> <span m="2863830">of</span> <span m="2864010">n</span> <span m="2864170">horses.</span> <span m="2868430">All</span> <span m="2868510">right.</span> <span m="2871170">Well,</span> <span m="2874700">since</span> <span m="2877020">the</span> <span m="2877160">color</span> <span m="2877510">of</span> <span m="2877580">h1</span> <span m="2881292">equals</span> <span m="2881760">the</span> <span m="2881860">color</span> <span m="2882320">of</span> <span m="2882820">those</span> <span m="2883110">guys,</span> <span m="2886080">h2</span> <span m="2887242">to</span> <span m="2887640">hn,</span> <span m="2889430">I</span> <span m="2889540">know</span> <span m="2889770">h1</span> <span m="2890180">is</span> <span m="2890270">the</span> <span m="2890350">same</span> <span m="2890560">color</span> <span m="2890920">as these</span> <span m="2891160">guys.</span> <span m="2892560">I</span> <span m="2892700">also</span> <span m="2892930">know</span> <span m="2893060">that</span> <span m="2893190">hn</span> <span m="2893610">plus</span> <span m="2893830">1</span> <span m="2894120">is</span> <span m="2894210">the</span> <span m="2894300">same</span> <span m="2894560">color</span> <span m="2894870">as</span> <span m="2894980">those</span> <span m="2895210">guys.</span> </p>
<p><span m="2902990">That</span> <span m="2903220">means</span> <span m="2903520">that</span> <span m="2903990">h1</span> <span m="2904460">is the</span> <span m="2904540">same</span> <span m="2904800">color</span> <span m="2905060">as</span> <span m="2905170">hn</span> <span m="2905500">plus</span> <span m="2905750">1.</span> <span m="2907440">And</span> <span m="2907640">all</span> <span m="2907840">n</span> <span m="2907970">plus</span> <span m="2908210">1</span> <span m="2908420">are</span> <span m="2908460">the</span> <span m="2908540">same</span> <span m="2908790">color.</span> <span m="2918410">And</span> <span m="2918620">that's</span> <span m="2918820">pn</span> <span m="2919120">plus</span> <span m="2919330">1.</span> <span m="2922460">That</span> <span m="2922710">implies</span> <span m="2923140">pn</span> <span m="2923390">plus</span> <span m="2923610">1.</span> <span m="2925660">And</span> <span m="2925880">I'm</span> <span m="2925980">done.</span> <span m="2929820">Now</span> <span m="2930760">a</span> <span m="2930820">few</span> <span m="2931040">years</span> <span m="2931400">ago</span> <span m="2932070">we</span> <span m="2932270">assigned</span> <span m="2932630">this</span> <span m="2932790">problem</span> <span m="2933120">as</span> <span m="2933240">homework.</span> <span m="2934720">And</span> <span m="2934930">we</span> <span m="2935060">asked</span> <span m="2935260">students</span> <span m="2935620">to</span> <span m="2935720">figure</span> <span m="2936050">out</span> <span m="2936210">what</span> <span m="2936410">went</span> <span m="2936610">wrong</span> <span m="2936980">with</span> <span m="2937120">the</span> <span m="2937200">problem.</span> <span m="2937860">Why</span> <span m="2938490">doesn't</span> <span m="2938740">this</span> <span m="2938870">work?</span> </p>
<p><span m="2939740">And</span> <span m="2940280">the</span> <span m="2940560">responses</span> <span m="2941160">were</span> <span m="2941310">a</span> <span m="2941360">little</span> <span m="2941700">discouraging.</span> <span m="2943230">Half</span> <span m="2943610">the</span> <span m="2943710">class</span> <span m="2944190">responded,</span> <span m="2945310">effectively</span> <span m="2946020">as</span> <span m="2946140">follows,</span> <span m="2946950">this</span> <span m="2947170">example</span> <span m="2947650">just</span> <span m="2947960">goes</span> <span m="2948180">to</span> <span m="2948270">show</span> <span m="2948590">that</span> <span m="2948740">induction</span> <span m="2949240">doesn't</span> <span m="2949590">always</span> <span m="2949980">work.</span> <span m="2952560">A</span> <span m="2952670">third</span> <span m="2952950">of</span> <span m="2953000">the</span> <span m="2953100">class</span> <span m="2954080">said,</span> <span m="2955010">I</span> <span m="2955060">always</span> <span m="2955430">knew</span> <span m="2955660">that you</span> <span m="2955850">can't</span> <span m="2956160">trust</span> <span m="2956470">mathematics.</span> <span m="2957390">This</span> <span m="2957680">example</span> <span m="2958210">just</span> <span m="2958780">proves</span> <span m="2959310">it.</span> <span m="2961140">That</span> <span m="2961310">really</span> <span m="2961530">hurt.</span> </p>
<p><span m="2963540">And</span> <span m="2963920">most</span> <span m="2964210">of</span> <span m="2964250">the</span> <span m="2964340">rest</span> <span m="2964640">were</span> <span m="2964800">something</span> <span m="2965260">similar</span> <span m="2965680">to</span> <span m="2965790">that.</span> <span m="2966930">Not</span> <span m="2967090">exactly</span> <span m="2967580">what</span> <span m="2967720">we</span> <span m="2967840">were</span> <span m="2967940">looking</span> <span m="2968260">for</span> <span m="2968650">in</span> <span m="2968750">the</span> <span m="2968820">homework.</span> <span m="2969180">So</span> <span m="2969300">now</span> <span m="2969560">we</span> <span m="2969940">don't</span> <span m="2970300">leave</span> <span m="2970480">it</span> <span m="2970570">to</span> <span m="2970640">homework.</span> <span m="2971030">We</span> <span m="2971300">do</span> <span m="2971440">it</span> <span m="2971500">in</span> <span m="2971590">class.</span> <span m="2974110">What's</span> <span m="2974490">the</span> <span m="2974600">flaw</span> <span m="2975000">here?</span> <span m="2977150">What was</span> <span m="2977360">it?</span> </p>
<p><span m="2978248">AUDIENCE: P of n.</span> </p>
<p><span m="2979580">PROFESSOR: P of n?</span> <span m="2979810">What's</span> <span m="2979960">wrong</span> <span m="2980150">with</span> <span m="2980300">p</span> <span m="2980430">of n?</span> </p>
<p><span m="2981279">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="2985690">PROFESSOR: No,</span> <span m="2986190">this</span> <span m="2986430">is</span> <span m="2986510">a</span> <span m="2986580">real</span> <span m="2986900">assumption</span> <span m="2987380">in</span> <span m="2987590">any</span> <span m="2987910">set</span> <span m="2988340">of</span> <span m="2988480">n</span> <span m="2988680">horses--</span> <span m="2990600">depends</span> <span m="2990880">on</span> <span m="2991020">n--</span> <span m="2991650">the</span> <span m="2991750">horses</span> <span m="2992170">are</span> <span m="2992280">all</span> <span m="2992420">the</span> <span m="2992500">same</span> <span m="2992750">color.</span> <span m="2993150">That</span> <span m="2993400">is</span> <span m="2993670">a</span> <span m="2994430">proposition.</span> <span m="2995730">Because</span> <span m="2995960">it</span> <span m="2996020">depends</span> <span m="2996270">on</span> <span m="2996400">n,</span> <span m="2996510">it's</span> <span m="2996630">a</span> <span m="2996680">predicate.</span> <span m="2998340">It's</span> <span m="2998530">true</span> <span m="2998870">or it's</span> <span m="2999040">false.</span> <span m="3000990">Now</span> <span m="3001470">we</span> <span m="3001590">know</span> <span m="3001790">it's</span> <span m="3001920">false,</span> <span m="3003030">because</span> <span m="3003200">you</span> <span m="3003310">could</span> <span m="3003480">get</span> <span m="3003620">a</span> <span m="3003690">set</span> <span m="3003890">of</span> <span m="3004285">a</span> <span m="3004550">couple</span> <span m="3004840">horses</span> <span m="3005220">with</span> <span m="3005350">different</span> <span m="3005600">color.</span> <span m="3006670">But</span> <span m="3006820">it</span> <span m="3006900">is</span> <span m="3007050">a</span> <span m="3007100">proposition.</span> </p>
<p><span m="3007740">Yeah,</span> <span m="3007940">way</span> <span m="3008050">back</span> <span m="3008250">there?</span> </p>
<p><span m="3010640">AUDIENCE: [INAUDIBLE]</span> <span m="3011450">for like a</span> <span m="3011780">certain</span> <span m="3012200">set of</span> <span m="3012550">horses.</span> <span m="3013840">So</span> <span m="3014260">even</span> <span m="3014680">though it's</span> <span m="3015100">the same number</span> <span m="3015520">of horses,</span> <span m="3015940">so it</span> <span m="3016320">was</span> <span m="3016785">the</span> <span m="3017250">same</span> <span m="3018686">number as</span> <span m="3019174">a different</span> <span m="3019662">set of</span> <span m="3020150">horses</span> <span m="3020638">it's</span> <span m="3021126">not something</span> <span m="3021614">you can</span> <span m="3022102">assume</span> <span m="3022590">anymore.</span> </p>
<p><span m="3023570">PROFESSOR: That's</span> <span m="3023800">a</span> <span m="3023860">great</span> <span m="3024190">point.</span> <span m="3024410">I</span> <span m="3024480">did</span> <span m="3024830">gloss</span> <span m="3025240">over</span> <span m="3025570">something</span> <span m="3025970">here,</span> <span m="3026870">because</span> <span m="3027400">in</span> <span m="3027660">the</span> <span m="3027760">predicate</span> <span m="3028450">I've</span> <span m="3028610">got</span> <span m="3029310">"in</span> <span m="3029660">any</span> <span m="3029990">set."</span> <span m="3030860">So</span> <span m="3031050">there's</span> <span m="3031230">really</span> <span m="3031740">a</span> <span m="3032210">"for</span> <span m="3032680">all</span> <span m="3033010">sets"</span> <span m="3033660">sitting</span> <span m="3033980">out</span> <span m="3034130">here.</span> <span m="3035040">So</span> <span m="3035060">I've</span> <span m="3035150">going</span> <span m="3035280">to</span> <span m="3035330">be</span> <span m="3035440">careful</span> <span m="3035890">that</span> <span m="3036000">I</span> <span m="3036140">establish</span> <span m="3036810">that</span> <span m="3037230">when</span> <span m="3037360">I'm</span> <span m="3038230">trying</span> <span m="3038530">to</span> <span m="3038700">prove</span> <span m="3038960">pn</span> <span m="3039930">implies</span> <span m="3040350">pn</span> <span m="3040590">plus</span> <span m="3040800">1.</span> <span m="3041080">So</span> <span m="3042040">to</span> <span m="3042160">establish</span> <span m="3042640">pn</span> <span m="3042910">plus</span> <span m="3043110">1</span> <span m="3043330">here,</span> <span m="3043450">I</span> <span m="3043520">assume</span> <span m="3043810">pn</span> <span m="3044160">which</span> <span m="3044310">means</span> <span m="3044540">in</span> <span m="3044660">any</span> <span m="3044960">set</span> <span m="3045180">of</span> <span m="3045260">n</span> <span m="3045370">horses</span> <span m="3045710">they're</span> <span m="3046050">all</span> <span m="3046220">the</span> <span m="3046340">same</span> <span m="3046530">color.</span> <span m="3046860">That</span> <span m="3047060">I'm</span> <span m="3047190">given.</span> <span m="3047910">That's</span> <span m="3048100">pn.</span> </p>
<p><span m="3048840">I've</span> <span m="3049210">got</span> <span m="3049330">to</span> <span m="3049400">look</span> <span m="3049590">at</span> <span m="3049790">any</span> <span m="3050270">set</span> <span m="3051300">of</span> <span m="3051440">n</span> <span m="3051580">plus</span> <span m="3051800">1</span> <span m="3051980">horses.</span> <span m="3052500">So</span> <span m="3052650">consider</span> <span m="3053590">any</span> <span m="3053880">set,</span> <span m="3055180">not</span> <span m="3055380">a</span> <span m="3055550">set,</span> <span m="3055830">any</span> <span m="3056110">set</span> <span m="3056910">of</span> <span m="3057030">n</span> <span m="3057160">plus</span> <span m="3057370">1</span> <span m="3057530">horses.</span> <span m="3058270">So you</span> <span m="3058470">pick</span> <span m="3058700">any</span> <span m="3058870">set</span> <span m="3059080">you</span> <span m="3059180">want.</span> <span m="3060060">Call them</span> <span m="3060470">this,</span> <span m="3061600">h1</span> <span m="3062050">through</span> <span m="3062280">hn plus</span> <span m="3062780">1.</span> <span m="3063870">Well</span> <span m="3064420">then,</span> <span m="3064820">the</span> <span m="3064910">first</span> <span m="3065260">n</span> <span m="3065740">are</span> <span m="3066200">a</span> <span m="3066310">set</span> <span m="3066660">of</span> <span m="3066810">n</span> <span m="3066950">horses,</span> <span m="3067420">so</span> <span m="3067530">I</span> <span m="3067590">can</span> <span m="3067720">apply</span> <span m="3068030">pn,</span> <span m="3068900">therefore</span> <span m="3069220">they</span> <span m="3069330">have</span> <span m="3069510">the</span> <span m="3069580">same</span> <span m="3069810">color.</span> </p>
<p><span m="3071290">The</span> <span m="3071410">last</span> <span m="3071840">n</span> <span m="3072060">horses</span> <span m="3072450">in</span> <span m="3072500">the</span> <span m="3072580">set</span> <span m="3073240">are</span> <span m="3073470">a</span> <span m="3073520">set</span> <span m="3073720">of n</span> <span m="3073900">horses,</span> <span m="3074860">so</span> <span m="3075010">I</span> <span m="3075070">can imply</span> <span m="3075460">pn.</span> <span m="3076590">So</span> <span m="3076770">these</span> <span m="3077000">guys</span> <span m="3077260">all</span> <span m="3077490">have</span> <span m="3077630">the</span> <span m="3077700">same</span> <span m="3077940">color,</span> <span m="3078570">and</span> <span m="3078790">I'm</span> <span m="3078980">on</span> <span m="3079070">my</span> <span m="3079180">merry</span> <span m="3079490">way</span> <span m="3079750">here.</span> <span m="3080810">So</span> <span m="3080940">you're</span> <span m="3081120">right,</span> <span m="3081360">I</span> <span m="3081420">had</span> <span m="3081690">to</span> <span m="3081780">do</span> <span m="3082000">a</span> <span m="3082060">little</span> <span m="3082250">bit</span> <span m="3082380">more</span> <span m="3082620">work,</span> <span m="3083800">but</span> <span m="3083930">I</span> <span m="3083990">can</span> <span m="3084150">do</span> <span m="3084280">that</span> <span m="3084480">work</span> <span m="3085080">and</span> <span m="3085160">I</span> <span m="3085220">still</span> <span m="3085440">get</span> <span m="3085560">a proof.</span> <span m="3086660">Yeah?</span> </p>
<p><span m="3088114">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="3089560">PROFESSOR: Good</span> <span m="3089720">question.</span> <span m="3089970">Is</span> <span m="3090080">there a</span> <span m="3090140">problem</span> <span m="3090430">with</span> <span m="3090480">the</span> <span m="3090560">base</span> <span m="3090980">case?</span> <span m="3091950">So</span> <span m="3092140">my</span> <span m="3092270">base</span> <span m="3092640">case</span> <span m="3093180">was</span> <span m="3094110">here,</span> <span m="3095020">n</span> <span m="3095210">equals</span> <span m="3095500">1.</span> <span m="3097270">In</span> <span m="3097480">any</span> <span m="3097740">set</span> <span m="3098030">of</span> <span m="3098160">one</span> <span m="3098440">horse,</span> <span m="3098920">the</span> <span m="3099010">horses--</span> <span m="3100490">let's</span> <span m="3100700">say</span> <span m="3101380">in</span> <span m="3101630">the</span> <span m="3101720">set</span> <span m="3102060">just</span> <span m="3102150">to</span> <span m="3102230">be</span> <span m="3102330">really</span> <span m="3104350">careful--</span> <span m="3106450">are</span> <span m="3106620">all</span> <span m="3106800">the</span> <span m="3106860">same</span> <span m="3107110">color.</span> <span m="3107960">No,</span> <span m="3108180">in</span> <span m="3108290">any</span> <span m="3108490">set</span> <span m="3108720">of</span> <span m="3108810">one</span> <span m="3109020">horses</span> <span m="3110430">there's</span> <span m="3110580">only</span> <span m="3110730">one</span> <span m="3110880">horse,</span> <span m="3111100">so</span> <span m="3111310">it's</span> <span m="3111450">the</span> <span m="3111540">same</span> <span m="3111780">color as</span> <span m="3112170">itself.</span> <span m="3113080">Yeah?</span> </p>
<p><span m="3114504">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="3117780">PROFESSOR: n</span> <span m="3118180">plus</span> <span m="3118350">1</span> <span m="3118480">is</span> <span m="3118580">not</span> <span m="3118840">the</span> <span m="3118930">same</span> <span m="3119200">as n,</span> <span m="3119674">yeah.</span> </p>
<p><span m="3120622">AUDIENCE: [INAUDIBLE].</span> <span m="3122088">So you</span> <span m="3122556">can't</span> <span m="3123024">have the</span> <span m="3123492">same assumption</span> <span m="3124896">because it's</span> <span m="3125364">not</span> <span m="3125832">a set</span> <span m="3126300">of n horses.</span> <span m="3126768">It's</span> <span m="3127236">a set of n</span> <span m="3127704">plus 1?</span> </p>
<p><span m="3128650">PROFESSOR: Well,</span> <span m="3129150">let's</span> <span m="3129380">see.</span> <span m="3129520">So</span> <span m="3129820">you're</span> <span m="3130090">arguing</span> <span m="3130360">I</span> <span m="3130390">made</span> <span m="3130560">a</span> <span m="3130590">mistake</span> <span m="3130930">here,</span> <span m="3131270">right?</span> <span m="3132700">Well,</span> <span m="3133710">I've</span> <span m="3133880">got</span> <span m="3134040">horses--</span> <span m="3134630">horse</span> <span m="3134870">2,</span> <span m="3135190">3,</span> <span m="3135510">up</span> <span m="3135590">to</span> <span m="3135670">n</span> <span m="3135810">plus</span> <span m="3136020">1.</span> <span m="3136230">How</span> <span m="3136400">many</span> <span m="3136550">horses</span> <span m="3137070">are</span> <span m="3137190">in</span> <span m="3137310">this</span> <span m="3137530">set</span> <span m="3137750">here?</span> </p>
<p><span m="3140640">AUDIENCE: Oh,</span> <span m="3140915">in</span> <span m="3141190">that</span> <span m="3141420">set?</span> </p>
<p><span m="3141820">PROFESSOR: In</span> <span m="3142030">this</span> <span m="3142330">set.</span> <span m="3142760">How many horses</span> <span m="3143160">are</span> <span m="3143280">there?</span> </p>
<p><span m="3144540">AUDIENCE: N.</span> </p>
<p><span m="3144900">PROFESSOR: N.</span> <span m="3145590">So</span> <span m="3145740">I</span> <span m="3145810">can</span> <span m="3145940">apply</span> <span m="3146340">p</span> <span m="3146630">of</span> <span m="3146710">n</span> <span m="3146890">to</span> <span m="3146990">this</span> <span m="3147390">set</span> <span m="3147630">just</span> <span m="3147930">the</span> <span m="3148020">same</span> <span m="3148270">as</span> <span m="3148360">that</span> <span m="3148570">set,</span> <span m="3148760">because</span> <span m="3148870">p</span> <span m="3149010">of</span> <span m="3149070">n</span> <span m="3149180">is</span> <span m="3149570">any</span> <span m="3149880">set</span> <span m="3150180">of n horses.</span> <span m="3151290">Well,</span> <span m="3151490">I'm</span> <span m="3151660">picking</span> <span m="3151930">this</span> <span m="3152100">one</span> <span m="3152260">now</span> <span m="3152935">and</span> <span m="3153310">applying</span> <span m="3153720">pn</span> <span m="3154060">to</span> <span m="3154240">it.</span> <span m="3155410">And</span> <span m="3155560">so</span> <span m="3155670">therefore</span> <span m="3155990">they're</span> <span m="3156150">all</span> <span m="3156280">the</span> <span m="3156340">same</span> <span m="3156540">color.</span> <span m="3159760">Yeah?</span> </p>
<p><span m="3161646">AUDIENCE: H1,</span> <span m="3162128">h2,</span> <span m="3162610">dot, dot,</span> <span m="3163092">dot.</span> </p>
<p><span m="3164060">PROFESSOR: Yes.</span> </p>
<p><span m="3165450">AUDIENCE: Because</span> <span m="3165944">it does</span> <span m="3166438">work</span> <span m="3166932">it there's</span> <span m="3167426">two</span> <span m="3167920">or more,</span> <span m="3168414">but you didn't</span> <span m="3168908">prove</span> <span m="3169402">it with just</span> <span m="3169896">one--</span> <span m="3170390">or from</span> <span m="3170884">one to</span> <span m="3171378">two.</span> </p>
<p><span m="3172370">PROFESSOR: Exactly.</span> <span m="3173175">Remember</span> <span m="3173540">I told you</span> <span m="3173590">that</span> <span m="3173810">dot,</span> <span m="3174070">dot,</span> <span m="3174320">dot</span> <span m="3174585">is</span> <span m="3174850">so</span> <span m="3175020">reasonable,</span> <span m="3176700">so</span> <span m="3176900">easy</span> <span m="3177210">to</span> <span m="3177260">use.</span> <span m="3177740">Everybody</span> <span m="3178140">uses</span> <span m="3178440">it.</span> <span m="3178540">It was</span> <span m="3178670">going</span> <span m="3178790">to</span> <span m="3179260">catch</span> <span m="3179560">us</span> <span m="3179700">up.</span> <span m="3180440">The</span> <span m="3180550">bug</span> <span m="3180830">is in the</span> <span m="3181270">dot,</span> <span m="3181540">dot,</span> <span m="3182010">dot.</span> <span m="3182420">And</span> <span m="3182570">in</span> <span m="3182640">particular,</span> <span m="3183310">it</span> <span m="3184340">has</span> <span m="3184630">to</span> <span m="3184720">do</span> <span m="3184950">with</span> <span m="3185080">the</span> <span m="3185180">case</span> <span m="3186350">n</span> <span m="3187280">equal</span> <span m="3188740">2.</span> <span m="3189150">So</span> <span m="3189260">there's</span> <span m="3189390">two</span> <span m="3189540">ways</span> <span m="3189770">to</span> <span m="3189820">look</span> <span m="3189960">at</span> <span m="3190010">the</span> <span m="3190090">bug,</span> <span m="3190400">dot,</span> <span m="3190610">dot,</span> <span m="3190800">dot,</span> <span m="3191480">or</span> <span m="3191840">we</span> <span m="3191990">didn't</span> <span m="3192710">completely</span> <span m="3193580">do</span> <span m="3193780">all</span> <span m="3193950">the</span> <span m="3194070">inductive</span> <span m="3194500">steps.</span> </p>
<p><span m="3196090">So</span> <span m="3196270">let's</span> <span m="3196610">look</span> <span m="3196830">at</span> <span m="3196900">the</span> <span m="3197010">case--</span> <span m="3198680">actually</span> <span m="3199030">it's</span> <span m="3199160">the</span> <span m="3199210">case</span> <span m="3199540">n</span> <span m="3199680">equals</span> <span m="3199950">1,</span> <span m="3202800">here,</span> <span m="3203500">in</span> <span m="3203630">this</span> <span m="3203820">inductive</span> <span m="3204210">step.</span> <span m="3205090">Did</span> <span m="3205300">I</span> <span m="3205400">prove</span> <span m="3206590">p1</span> <span m="3207580">implies</span> <span m="3208220">p2?</span> <span m="3210780">Let's</span> <span m="3211180">just</span> <span m="3211460">double</span> <span m="3211720">check</span> <span m="3212000">that</span> <span m="3212170">worked</span> <span m="3212430">for</span> <span m="3212550">n</span> <span m="3212630">equals</span> <span m="3212880">1.</span> <span m="3220630">n</span> <span m="3220770">equals</span> <span m="3221120">1.</span> <span m="3222066">I've</span> <span m="3222540">got</span> <span m="3222740">a</span> <span m="3222800">set</span> <span m="3223650">of</span> <span m="3224900">two</span> <span m="3225290">horses.</span> <span m="3226030">This</span> <span m="3226280">becomes</span> <span m="3226680">2.</span> <span m="3228360">So</span> <span m="3228470">I've</span> <span m="3228570">got</span> <span m="3228820">h1,</span> <span m="3229380">h2.</span> <span m="3231640">Nothing</span> <span m="3231950">in</span> <span m="3232030">the</span> <span m="3232110">dot,</span> <span m="3232320">dot,</span> <span m="3232520">dot.</span> <span m="3232750">It's</span> <span m="3232860">just</span> <span m="3233060">h1</span> <span m="3233240">and</span> <span m="3233480">h2.</span> </p>
<p><span m="3235310">Then</span> <span m="3237330">this</span> <span m="3237780">becomes</span> <span m="3239410">h1.</span> <span m="3241570">So</span> <span m="3241730">sure</span> <span m="3241950">enough,</span> <span m="3242180">h1</span> <span m="3242650">is</span> <span m="3242750">the</span> <span m="3242830">same</span> <span m="3243060">color as</span> <span m="3243400">itself.</span> <span m="3244260">This</span> <span m="3244630">becomes--</span> <span m="3245660">what</span> <span m="3245790">does</span> <span m="3245890">this</span> <span m="3246040">become,</span> <span m="3246360">this</span> <span m="3246450">set</span> <span m="3246630">of</span> <span m="3246720">horses?</span> <span m="3247170">What</span> <span m="3247220">is</span> <span m="3247320">it</span> <span m="3247400">really?</span> <span m="3248990">h2.</span> <span m="3251590">All</span> <span m="3251770">right,</span> <span m="3252350">so</span> <span m="3252460">I've</span> <span m="3252540">got</span> <span m="3252700">the</span> <span m="3252780">color</span> <span m="3253120">of h1</span> <span m="3253190">equals</span> <span m="3253940">the</span> <span m="3254030">color</span> <span m="3254430">of--</span> <span m="3255940">oh</span> <span m="3256190">man,</span> <span m="3256380">this</span> <span m="3256550">is</span> <span m="3256630">so</span> <span m="3256810">hard</span> <span m="3257020">to</span> <span m="3257090">see</span> <span m="3257240">this</span> <span m="3257430">bug.</span> <span m="3258850">I</span> <span m="3258970">wrote</span> <span m="3259210">down</span> <span m="3259440">h2</span> <span m="3260170">dot,</span> <span m="3260350">dot,</span> <span m="3260490">dot,</span> <span m="3260690">to</span> <span m="3260760">hn</span> <span m="3261260">because--</span> <span m="3263140">take</span> <span m="3263400">out</span> <span m="3263550">h1,</span> <span m="3265510">what's</span> <span m="3265840">left?</span> <span m="3266410">What's</span> <span m="3266610">left</span> <span m="3266800">is</span> <span m="3266910">h2</span> <span m="3267070">through</span> <span m="3267220">hn.</span> </p>
<p><span m="3268550">But</span> <span m="3268700">this</span> <span m="3268850">set</span> <span m="3269030">is</span> <span m="3269150">only</span> <span m="3269390">h1.</span> <span m="3269870">What's</span> <span m="3270170">really</span> <span m="3270420">left</span> <span m="3270690">here?</span> <span m="3270830">What</span> <span m="3271000">is</span> <span m="3271150">this</span> <span m="3271280">set?</span> <span m="3274400">What</span> <span m="3274610">is</span> <span m="3274770">this</span> <span m="3274980">set?</span> <span m="3279980">h2,</span> <span m="3280620">h1?</span> <span m="3281962">No.</span> <span m="3283450">You</span> <span m="3283520">see,</span> <span m="3283910">what</span> <span m="3284100">is</span> <span m="3284220">the</span> <span m="3284310">whole</span> <span m="3284600">set?</span> <span m="3284830">h1,</span> <span m="3285040">dot,</span> <span m="3285300">dot,</span> <span m="3285560">dot,</span> <span m="3285750">hn.</span> <span m="3286110">What</span> <span m="3286240">is</span> <span m="3286320">the</span> <span m="3286400">whole</span> <span m="3286660">thing?</span> <span m="3286870">It's</span> <span m="3286960">just</span> <span m="3287190">h1.</span> <span m="3288930">pull</span> <span m="3289370">out h1</span> <span m="3289800">and</span> <span m="3289840">look</span> <span m="3290010">at</span> <span m="3290060">the</span> <span m="3290150">rest</span> <span m="3290420">of</span> <span m="3290510">it,</span> <span m="3292950">how</span> <span m="3293110">many</span> <span m="3293250">horses</span> <span m="3293690">are</span> <span m="3294000">here?</span> <span m="3295600">0</span> <span m="3296240">horses.</span> </p>
<p><span m="3299240">This</span> <span m="3299730">is</span> <span m="3299880">the</span> <span m="3300070">empty</span> <span m="3300350">set.</span> <span m="3301120">There</span> <span m="3301310">are</span> <span m="3301430">no</span> <span m="3301790">horses</span> <span m="3302280">here</span> <span m="3302460">to</span> <span m="3302530">compare</span> <span m="3302930">to.</span> <span m="3304560">Even</span> <span m="3304770">though</span> <span m="3304840">I</span> <span m="3304920">got</span> <span m="3305150">an h2,</span> <span m="3306130">I</span> <span m="3306200">got</span> <span m="3306340">an</span> <span m="3306430">hn,</span> <span m="3306775">I got a</span> <span m="3307120">dot, dot,</span> <span m="3307320">dot.</span> <span m="3308220">Because</span> <span m="3309050">generally,</span> <span m="3309510">if n is</span> <span m="3309880">big,</span> <span m="3311550">this</span> <span m="3311770">has</span> <span m="3312350">n</span> <span m="3312550">minus</span> <span m="3312860">1</span> <span m="3313010">horses</span> <span m="3313410">are</span> <span m="3313490">here.</span> <span m="3313870">There's</span> <span m="3314060">n</span> <span m="3314330">total.</span> <span m="3315310">I</span> <span m="3315420">got</span> <span m="3315590">n</span> <span m="3315720">minus</span> <span m="3316030">1</span> <span m="3316200">right</span> <span m="3316460">here.</span> <span m="3318120">But</span> <span m="3318240">n</span> <span m="3318360">minus</span> <span m="3318630">1</span> <span m="3318860">is</span> <span m="3319080">0.</span> <span m="3320280">There</span> <span m="3320490">are</span> <span m="3320600">no</span> <span m="3321030">horses</span> <span m="3321450">here.</span> <span m="3322680">And</span> <span m="3322880">so this</span> <span m="3323160">bridge</span> <span m="3323530">in the</span> <span m="3323730">equality</span> <span m="3324200">totally</span> <span m="3324550">breaks.</span> </p>
<p><span m="3327220">I</span> <span m="3327440">got</span> <span m="3327690">color</span> <span m="3327990">of</span> <span m="3328060">h1 equals--</span> <span m="3330240">there's</span> <span m="3330440">no</span> <span m="3330640">information</span> <span m="3331130">here--</span> <span m="3331910">equals</span> <span m="3332200">the</span> <span m="3332290">color</span> <span m="3332770">of</span> <span m="3332900">h2.</span> <span m="3334040">There's</span> <span m="3334210">no</span> <span m="3334440">equality</span> <span m="3335050">here,</span> <span m="3335260">because</span> <span m="3335430">there's</span> <span m="3335590">no</span> <span m="3335750">horses</span> <span m="3336130">in</span> <span m="3336210">this</span> <span m="3336440">set,</span> <span m="3337660">all</span> <span m="3337850">because</span> <span m="3338120">of</span> <span m="3338160">that</span> <span m="3338330">dot,</span> <span m="3338550">dot,</span> <span m="3338750">dot.</span> <span m="3339380">Do you</span> <span m="3339520">see</span> <span m="3339790">where</span> <span m="3339920">the</span> <span m="3340300">problem</span> <span m="3340700">is</span> <span m="3341070">by</span> <span m="3341160">using</span> <span m="3341360">the</span> <span m="3341440">dot,</span> <span m="3341640">dot,</span> <span m="3341830">dot?</span> </p>
<p><span m="3343360">For</span> <span m="3343440">the</span> <span m="3343530">case</span> <span m="3343790">n</span> <span m="3343940">equals--</span> <span m="3344320">it</span> <span m="3344390">was</span> <span m="3344510">true</span> <span m="3344900">for</span> <span m="3345090">every</span> <span m="3345450">other</span> <span m="3345620">case</span> <span m="3345980">of</span> <span m="3346080">n.</span> <span m="3346912">n</span> <span m="3347330">equals</span> <span m="3347570">2,</span> <span m="3347780">n</span> <span m="3347870">equals</span> <span m="3348090">3,</span> <span m="3348290">it's</span> <span m="3348450">all</span> <span m="3348630">true.</span> <span m="3351240">In</span> <span m="3351340">fact,</span> <span m="3352410">what</span> <span m="3352470">we</span> <span m="3352610">proved,</span> <span m="3353240">we</span> <span m="3353420">proved</span> <span m="3353670">the</span> <span m="3353730">base</span> <span m="3354070">case</span> <span m="3354460">of</span> <span m="3354630">p1</span> <span m="3355360">is</span> <span m="3355450">true.</span> <span m="3357030">This</span> <span m="3357420">is</span> <span m="3357520">an</span> <span m="3357610">argument</span> <span m="3358000">that</span> <span m="3358090">p2</span> <span m="3358640">implies</span> <span m="3359030">p3.</span> <span m="3359610">That</span> <span m="3359780">is</span> <span m="3359900">true.</span> <span m="3361790">p3</span> <span m="3363890">implies</span> <span m="3364340">p4,</span> <span m="3365120">that</span> <span m="3365310">is</span> <span m="3365420">true.</span> <span m="3366680">And</span> <span m="3366800">so</span> <span m="3367010">forth.</span> </p>
<p><span m="3368200">We</span> <span m="3368270">proved</span> <span m="3368690">for</span> <span m="3368860">all</span> <span m="3369160">n</span> <span m="3369400">bigger</span> <span m="3369670">than or</span> <span m="3369850">equal</span> <span m="3370070">to</span> <span m="3370160">2,</span> <span m="3371050">pn</span> <span m="3372330">implies</span> <span m="3372830">pn</span> <span m="3373100">plus</span> <span m="3373360">1.</span> <span m="3373780">That</span> <span m="3373970">we</span> <span m="3374100">proved.</span> <span m="3375440">What</span> <span m="3375610">is</span> <span m="3375720">the</span> <span m="3375830">one</span> <span m="3376220">missing</span> <span m="3376630">implication</span> <span m="3377210">we</span> <span m="3377290">did</span> <span m="3377350">not</span> <span m="3377590">prove?</span> <span m="3380060">The</span> <span m="3380140">missing</span> <span m="3380690">link,</span> <span m="3381230">p1</span> <span m="3381780">implies</span> <span m="3382230">p2.</span> <span m="3384410">We</span> <span m="3384560">did</span> <span m="3384770">not</span> <span m="3385130">prove</span> <span m="3385390">that.</span> <span m="3386080">Now</span> <span m="3386250">is</span> <span m="3386440">that</span> <span m="3386660">true?</span> <span m="3388650">No.</span> <span m="3390350">You can</span> <span m="3390500">find</span> <span m="3390770">a</span> <span m="3390820">set</span> <span m="3390980">of</span> <span m="3391060">two</span> <span m="3391230">horses</span> <span m="3391610">are</span> <span m="3391700">not</span> <span m="3391870">the</span> <span m="3391930">same</span> <span m="3392170">color.</span> <span m="3393330">And</span> <span m="3393450">just</span> <span m="3393630">because</span> <span m="3393910">every</span> <span m="3394440">horse</span> <span m="3394800">is</span> <span m="3394900">the</span> <span m="3394970">same</span> <span m="3395180">color</span> <span m="3395480">as itself,</span> <span m="3395880">does</span> <span m="3396010">not</span> <span m="3396370">give</span> <span m="3396520">you</span> <span m="3396640">that.</span> </p>
<p><span m="3397400">That</span> <span m="3397780">was</span> <span m="3397900">missing</span> <span m="3398350">in</span> <span m="3398430">this</span> <span m="3398600">proof.</span> <span m="3399470">And</span> <span m="3399610">so</span> <span m="3399680">we</span> <span m="3399750">have</span> <span m="3399880">to</span> <span m="3399940">be</span> <span m="3400050">really</span> <span m="3400460">careful</span> <span m="3400890">when</span> <span m="3400980">you're</span> <span m="3401070">doing</span> <span m="3401320">these</span> <span m="3401510">proofs</span> <span m="3402060">that</span> <span m="3402760">you</span> <span m="3402960">establish</span> <span m="3403510">the</span> <span m="3403620">inductive</span> <span m="3404100">step</span> <span m="3404460">for</span> <span m="3404600">all</span> <span m="3405070">n</span> <span m="3405380">bigger</span> <span m="3405620">than</span> <span m="3405770">or equal</span> <span m="3405960">to the</span> <span m="3406070">base</span> <span m="3406340">case.</span> <span m="3407540">And</span> <span m="3407670">then</span> <span m="3407820">make</span> <span m="3408000">sure,</span> <span m="3408250">if</span> <span m="3408300">you're</span> <span m="3408390">going</span> <span m="3408490">to</span> <span m="3408530">use</span> <span m="3408800">this</span> <span m="3408980">really</span> <span m="3409340">convenient</span> <span m="3409800">dot</span> <span m="3409990">dot</span> <span m="3410160">dot</span> <span m="3410390">notation,</span> <span m="3411490">that you</span> <span m="3411790">don't</span> <span m="3411980">wind</span> <span m="3412260">up</span> <span m="3412410">here</span> <span m="3412760">saying,</span> <span m="3413140">oh</span> <span m="3413160">this</span> <span m="3413360">is</span> <span m="3413480">horses</span> <span m="3414040">h2</span> <span m="3414450">through</span> <span m="3414840">hn</span> <span m="3415650">when</span> <span m="3415820">there's</span> <span m="3415950">no</span> <span m="3416120">horses</span> <span m="3416490">there,</span> <span m="3418240">because</span> <span m="3418440">n is</span> <span m="3418610">1.</span> </p>
<p><span m="3420640">Any</span> <span m="3420830">questions</span> <span m="3421290">about</span> <span m="3421530">this?</span> <span m="3422370">Yeah?</span> </p>
<p><span m="3423688">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="3428580">PROFESSOR: Great</span> <span m="3428980">question.</span> <span m="3429670">All</span> <span m="3429930">right,</span> <span m="3430180">let's</span> <span m="3430400">fix</span> <span m="3430640">the</span> <span m="3430740">proof.</span> <span m="3431970">Start</span> <span m="3432320">with</span> <span m="3432410">the</span> <span m="3432490">base</span> <span m="3432870">case</span> <span m="3433150">of</span> <span m="3433260">p</span> <span m="3433490">of</span> <span m="3433570">2,</span> <span m="3434500">and</span> <span m="3434670">now</span> <span m="3434830">I've</span> <span m="3434960">got</span> <span m="3435100">all</span> <span m="3435500">this</span> <span m="3435750">done.</span> <span m="3436250">So</span> <span m="3436280">therefore,</span> <span m="3437430">that's</span> <span m="3437630">another</span> <span m="3437890">proof.</span> <span m="3439910">Yeah?</span> <span m="3443530">Say</span> <span m="3443700">it</span> <span m="3443760">again.</span> </p>
<p><span m="3444766">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="3449360">PROFESSOR: This</span> <span m="3449540">is</span> <span m="3449620">still</span> <span m="3449850">the</span> <span m="3449950">same.</span> <span m="3450250">In</span> <span m="3450360">any</span> <span m="3450670">set</span> <span m="3451450">of</span> <span m="3451660">n</span> <span m="3451980">bigger</span> <span m="3452090">than</span> <span m="3452240">or</span> <span m="3452280">equal</span> <span m="3452550">to</span> <span m="3452940">two</span> <span m="3453370">horses,</span> <span m="3454700">all</span> <span m="3454990">the</span> <span m="3455070">horses</span> <span m="3455460">in</span> <span m="3455520">the</span> <span m="3455610">set</span> <span m="3455800">are</span> <span m="3455860">the</span> <span m="3455940">same</span> <span m="3456150">color.</span> <span m="3456390">That's</span> <span m="3456580">what</span> <span m="3456660">he</span> <span m="3456750">saying.</span> <span m="3457390">And</span> <span m="3457520">he's</span> <span m="3457660">saying,</span> <span m="3457910">hey</span> <span m="3458090">look,</span> <span m="3458690">the</span> <span m="3458810">proof</span> <span m="3459070">worked</span> <span m="3459590">here.</span> <span m="3460000">The inductive</span> <span m="3460420">step</span> <span m="3460660">is</span> <span m="3460780">just</span> <span m="3461000">fine.</span> <span m="3461930">p2</span> <span m="3462750">does</span> <span m="3463020">imply</span> <span m="3463340">p3.</span> <span m="3464630">Yeah?</span> </p>
<p><span m="3465548">AUDIENCE: The base case</span> <span m="3465966">for 2</span> <span m="3466384">isn't</span> <span m="3466802">true.</span> </p>
<p><span m="3467220">PROFESSOR: That's</span> <span m="3467580">right.</span> <span m="3468120">The base</span> <span m="3468360">case</span> <span m="3468740">fails.</span> <span m="3471060">That's</span> <span m="3471180">why</span> <span m="3471300">you've</span> <span m="3471540">always</span> <span m="3472060">got</span> <span m="3472230">to</span> <span m="3472320">check</span> <span m="3472640">the</span> <span m="3472730">base</span> <span m="3473020">case.</span> <span m="3474450">Yeah?</span> </p>
<p><span m="3475390">AUDIENCE: What it</span> <span m="3475540">does</span> <span m="3476019">prove</span> <span m="3476498">is that</span> <span m="3477935">if you</span> <span m="3478414">find</span> <span m="3478893">any two</span> <span m="3479372">horse</span> <span m="3479851">and</span> <span m="3480330">they're</span> <span m="3480809">always</span> <span m="3481288">going to</span> <span m="3481767">be the same</span> <span m="3482246">color,</span> <span m="3482725">then all</span> <span m="3483204">horses have</span> <span m="3483683">to be the same</span> <span m="3484162">color.</span> </p>
<p><span m="3484650">PROFESSOR: That's</span> <span m="3484990">correct.</span> <span m="3485770">That's</span> <span m="3486000">a</span> <span m="3486050">great</span> <span m="3486310">point.</span> <span m="3487170">We</span> <span m="3487250">have</span> <span m="3487520">given</span> <span m="3487730">a</span> <span m="3487790">proof</span> <span m="3488250">that</span> <span m="3489050">if</span> <span m="3489220">you</span> <span m="3489350">look</span> <span m="3489520">at</span> <span m="3489600">any</span> <span m="3489870">pair</span> <span m="3490120">of</span> <span m="3490180">horses</span> <span m="3490580">and</span> <span m="3490660">they're</span> <span m="3490750">the</span> <span m="3490860">same</span> <span m="3491130">color,</span> <span m="3492190">then</span> <span m="3492360">all</span> <span m="3492700">horses</span> <span m="3493070">are</span> <span m="3493110">the</span> <span m="3493220">same</span> <span m="3493480">color.</span> <span m="3494380">That's</span> <span m="3494580">true.</span> <span m="3496620">That</span> <span m="3496820">is</span> <span m="3496900">true.</span> <span m="3497790">Of</span> <span m="3497880">course,</span> <span m="3498480">there</span> <span m="3498670">are</span> <span m="3498830">pairs</span> <span m="3499090">of</span> <span m="3499200">horses that</span> <span m="3499540">aren't</span> <span m="3499720">the</span> <span m="3499810">same</span> <span m="3500010">color.</span> <span m="3501000">So</span> <span m="3501130">the</span> <span m="3501220">base</span> <span m="3501490">case</span> <span m="3501720">would</span> <span m="3501860">fail.</span> </p>
<p><span m="3502560">So</span> <span m="3502840">always</span> <span m="3503370">check</span> <span m="3503610">the</span> <span m="3503700">base</span> <span m="3503980">case.</span> <span m="3504170">You</span> <span m="3504290">could</span> <span m="3504430">prove</span> <span m="3504660">some</span> <span m="3504810">great</span> <span m="3505280">stuff</span> <span m="3505670">if</span> <span m="3505780">you</span> <span m="3505860">don't</span> <span m="3506080">check</span> <span m="3506300">the</span> <span m="3506360">base</span> <span m="3506600">case.</span> <span m="3510670">All</span> <span m="3510930">right,</span> <span m="3511600">any</span> <span m="3512060">other</span> <span m="3512200">questions</span> <span m="3512590">about</span> <span m="3512840">that?</span> <span m="3514262">Yeah?</span> </p>
<p><span m="3517148">AUDIENCE: Negative</span> <span m="3517629">number</span> <span m="3518110">[INAUDIBLE]?</span> </p>
<p><span m="3520040">PROFESSOR: Yeah.</span> <span m="3520660">As</span> <span m="3521042">long as it's</span> <span m="3521424">an integer.</span> <span m="3521806">And as</span> <span m="3522190">long</span> <span m="3522420">as</span> <span m="3522510">you</span> <span m="3522810">prove</span> <span m="3523000">pn</span> <span m="3523395">implies</span> <span m="3523790">pn plus</span> <span m="3524070">1</span> <span m="3524260">starting</span> <span m="3524750">there</span> <span m="3524980">all</span> <span m="3525170">the</span> <span m="3525250">way</span> <span m="3525760">out.</span> <span m="3526390">Yeah,</span> <span m="3526400">you</span> <span m="3526530">can</span> <span m="3526640">start</span> <span m="3526830">at</span> <span m="3526880">negative</span> <span m="3527150">numbers</span> <span m="3527510">if</span> <span m="3527520">you</span> <span m="3527630">want.</span> <span m="3528990">Usually</span> <span m="3529520">there's</span> <span m="3529680">not</span> <span m="3529860">many</span> <span m="3530310">cases</span> <span m="3530710">where</span> <span m="3530820">that</span> <span m="3531100">comes</span> <span m="3531350">up</span> <span m="3531500">you</span> <span m="3531600">want</span> <span m="3531880">to,</span> <span m="3532000">but</span> <span m="3532150">you</span> <span m="3532260">can.</span> <span m="3532630">Nothing</span> <span m="3532870">wrong</span> <span m="3533100">with</span> <span m="3533190">that.</span> <span m="3535030">Any</span> <span m="3535210">other</span> <span m="3535340">questions?</span> </p>
<p><span m="3537730">Yeah,</span> <span m="3537860">you</span> <span m="3537950">can</span> <span m="3538070">see</span> <span m="3538290">why</span> <span m="3538450">it</span> <span m="3538530">was</span> <span m="3538690">a</span> <span m="3538750">messy</span> <span m="3539220">homework</span> <span m="3540140">problem.</span> <span m="3542820">So</span> <span m="3543030">far</span> <span m="3543200">we've</span> <span m="3543370">seen</span> <span m="3543540">examples</span> <span m="3544020">of</span> <span m="3544090">how</span> <span m="3544270">induction</span> <span m="3544790">is</span> <span m="3544900">useful</span> <span m="3546240">in</span> <span m="3546460">proving</span> <span m="3546900">the</span> <span m="3546970">hypothesis</span> <span m="3547660">is</span> <span m="3547760">true,</span> <span m="3548860">but</span> <span m="3548990">not</span> <span m="3549220">in</span> <span m="3549340">solving</span> <span m="3550060">the</span> <span m="3550160">problem,</span> <span m="3550910">per</span> <span m="3551080">se.</span> <span m="3552920">Or</span> <span m="3553180">even</span> <span m="3553380">figuring</span> <span m="3553800">out</span> <span m="3554030">what</span> <span m="3554210">the</span> <span m="3554280">hypothesis</span> <span m="3554920">should</span> <span m="3555120">be.</span> <span m="3556540">Now</span> <span m="3557490">in</span> <span m="3557650">the</span> <span m="3557750">last</span> <span m="3558050">example,</span> <span m="3559060">we're</span> <span m="3559300">going</span> <span m="3559400">to</span> <span m="3559460">show</span> <span m="3559760">you</span> <span m="3559950">how</span> <span m="3560130">induction</span> <span m="3560680">can</span> <span m="3560830">be</span> <span m="3560970">used</span> <span m="3561330">to</span> <span m="3561420">prove</span> <span m="3561910">there</span> <span m="3562110">is</span> <span m="3562390">a</span> <span m="3562460">solution</span> <span m="3562900">to</span> <span m="3562980">a</span> <span m="3563040">problem,</span> <span m="3564300">and</span> <span m="3564460">also</span> <span m="3564740">how</span> <span m="3564920">to</span> <span m="3565020">find</span> <span m="3565470">the</span> <span m="3565560">solution.</span> </p>
<p><span m="3566150">So</span> <span m="3566200">it's</span> <span m="3566320">actually</span> <span m="3566630">going</span> <span m="3566710">to</span> <span m="3566750">be</span> <span m="3566850">a</span> <span m="3566890">very</span> <span m="3567130">useful,</span> <span m="3567550">constructive</span> <span m="3568100">thing</span> <span m="3568310">in</span> <span m="3568370">this</span> <span m="3568540">case.</span> <span m="3569390">Now</span> <span m="3569540">this</span> <span m="3569750">problem</span> <span m="3570140">arose</span> <span m="3570640">in</span> <span m="3570730">the</span> <span m="3570820">construction</span> <span m="3571390">of</span> <span m="3571470">the</span> <span m="3571550">status</span> <span m="3571950">center,</span> <span m="3572840">the</span> <span m="3572900">building</span> <span m="3573410">we're</span> <span m="3573600">in</span> <span m="3573710">now.</span> <span m="3575110">This</span> <span m="3575480">whole</span> <span m="3575710">building</span> <span m="3576100">was</span> <span m="3576240">originally</span> <span m="3576800">supposed</span> <span m="3577140">to</span> <span m="3577210">cost,</span> <span m="3577950">completely</span> <span m="3578310">furnished,</span> <span m="3579100">under</span> <span m="3579350">$100</span> <span m="3579680">million.</span> <span m="3581120">That</span> <span m="3581300">was</span> <span m="3581430">the</span> <span m="3581490">goal.</span> </p>
<p><span m="3582400">But</span> <span m="3583040">the</span> <span m="3583120">first</span> <span m="3583440">mistake</span> <span m="3583820">they</span> <span m="3583930">made</span> <span m="3584310">was</span> <span m="3584440">the</span> <span m="3584530">first</span> <span m="3584780">step,</span> <span m="3585000">was</span> <span m="3585120">hiring</span> <span m="3585530">the</span> <span m="3585640">architect.</span> <span m="3586260">They</span> <span m="3586400">hired</span> <span m="3586730">Frank</span> <span m="3587020">Gehry.</span> <span m="3587820">I</span> <span m="3587900">think</span> <span m="3588220">MIT</span> <span m="3588600">is</span> <span m="3589090">now</span> <span m="3589260">in</span> <span m="3589310">a</span> <span m="3589350">lawsuit</span> <span m="3589610">with</span> <span m="3589870">Frank</span> <span m="3590100">Gehry.</span> <span m="3591350">So</span> <span m="3591480">he</span> <span m="3591630">was</span> <span m="3591750">the</span> <span m="3591880">architect.</span> <span m="3593200">And</span> <span m="3593610">costs</span> <span m="3594010">just</span> <span m="3594220">went</span> <span m="3594410">nuts.</span> <span m="3595600">As</span> <span m="3595660">you</span> <span m="3595760">could</span> <span m="3595870">imagine,</span> <span m="3596420">all</span> <span m="3596650">these</span> <span m="3596850">slanted</span> <span m="3597390">walls</span> <span m="3597960">and</span> <span m="3598250">crazy</span> <span m="3598570">things</span> <span m="3598820">happening</span> <span m="3599150">actually</span> <span m="3599600">are</span> <span m="3599670">expensive.</span> <span m="3600850">And</span> <span m="3602300">the</span> <span m="3602480">cost</span> <span m="3602830">quickly</span> <span m="3603140">got</span> <span m="3603310">over</span> <span m="3603480">$300</span> <span m="3603920">million,</span> <span m="3604190">literally.</span> </p>
<p><span m="3605940">That</span> <span m="3606230">that's</span> <span m="3606430">parts</span> <span m="3606700">true.</span> <span m="3606820">Now</span> <span m="3606940">I'm</span> <span m="3607080">going</span> <span m="3607170">to</span> <span m="3607540">fabricate</span> <span m="3608020">a</span> <span m="3608080">little</span> <span m="3608310">bit.</span> <span m="3610180">Now</span> <span m="3610390">actually,</span> <span m="3610700">fund</span> <span m="3611020">raising</span> <span m="3611320">became</span> <span m="3611700">a</span> <span m="3612100">huge</span> <span m="3612670">priority</span> <span m="3613590">once</span> <span m="3613810">they're</span> <span m="3614240">more</span> <span m="3614370">than</span> <span m="3614500">$200</span> <span m="3614780">million</span> <span m="3615020">over</span> <span m="3615180">budget</span> <span m="3615520">they</span> <span m="3615610">haven't</span> <span m="3616010">even bought</span> <span m="3616260">the</span> <span m="3616340">furniture</span> <span m="3616700">yet.</span> <span m="3617670">So</span> <span m="3618380">some</span> <span m="3618530">pretty</span> <span m="3618830">radical</span> <span m="3619290">ideas</span> <span m="3619770">were</span> <span m="3619870">proposed.</span> </p>
<p><span m="3621000">And</span> <span m="3621220">one</span> <span m="3621400">of</span> <span m="3621490">them</span> <span m="3622230">was</span> <span m="3622500">to</span> <span m="3622600">build</span> <span m="3622830">a</span> <span m="3622890">large</span> <span m="3623390">2</span> <span m="3623570">to</span> <span m="3623620">the</span> <span m="3623790">n</span> <span m="3624030">by</span> <span m="3624170">2</span> <span m="3624390">to the</span> <span m="3624550">n</span> <span m="3624740">courtyard</span> <span m="3626080">and</span> <span m="3626360">put</span> <span m="3626520">a</span> <span m="3626580">statue</span> <span m="3627080">of</span> <span m="3627180">a</span> <span m="3627250">wealthy,</span> <span m="3628260">potential</span> <span m="3628740">donor</span> <span m="3629750">in</span> <span m="3629880">the</span> <span m="3629960">center</span> <span m="3630400">of</span> <span m="3630450">the</span> <span m="3630550">courtyard.</span> <span m="3631990">So</span> <span m="3632300">let's</span> <span m="3634720">draw</span> <span m="3635020">this.</span> <span m="3647370">So</span> <span m="3647520">the</span> <span m="3647630">courtyard--</span> <span m="3650350">and</span> <span m="3650480">of</span> <span m="3650530">course,</span> <span m="3650780">you</span> <span m="3650860">know,</span> <span m="3650960">it's</span> <span m="3651100">computer</span> <span m="3651480">science,</span> <span m="3652050">so</span> <span m="3652070">it</span> <span m="3652110">has</span> <span m="3652350">to</span> <span m="3652420">be</span> <span m="3652510">a</span> <span m="3652560">power</span> <span m="3652870">of</span> <span m="3652970">2.</span> <span m="3655380">So</span> <span m="3655620">it's</span> <span m="3655760">2</span> <span m="3655940">to</span> <span m="3656000">the</span> <span m="3656170">n</span> <span m="3657170">by</span> <span m="3657435">2</span> <span m="3657700">to</span> <span m="3657900">the</span> <span m="3658080">n.</span> <span m="3659760">And</span> <span m="3659890">I've</span> <span m="3659970">drawn</span> <span m="3660270">here</span> <span m="3660440">the</span> <span m="3660550">case</span> <span m="3660880">n</span> <span m="3661000">equals</span> <span m="3661250">2.</span> </p>
<p><span m="3664060">And</span> <span m="3664230">we've</span> <span m="3664370">got</span> <span m="3664580">to</span> <span m="3664640">get</span> <span m="3666400">the</span> <span m="3666530">statue</span> <span m="3667040">of</span> <span m="3667120">the</span> <span m="3667210">wealthy</span> <span m="3667600">guy</span> <span m="3667960">in</span> <span m="3668070">the</span> <span m="3668150">middle.</span> <span m="3668760">And</span> <span m="3668930">I'm</span> <span m="3669000">not</span> <span m="3669180">supposed</span> <span m="3669450">to</span> <span m="3669510">reveal</span> <span m="3669800">his</span> <span m="3669990">name.</span> <span m="3670280">So</span> <span m="3670460">we're</span> <span m="3670570">just</span> <span m="3670740">going</span> <span m="3670830">to</span> <span m="3670890">call</span> <span m="3671140">him</span> <span m="3671520">Bill.</span> <span m="3675980">So</span> <span m="3676150">Bill's</span> <span m="3676480">got</span> <span m="3676630">to</span> <span m="3676680">go</span> <span m="3676850">in</span> <span m="3676910">the</span> <span m="3677000">center.</span> <span m="3678130">Now</span> <span m="3678380">this</span> <span m="3678590">would</span> <span m="3678760">be</span> <span m="3678900">fine,</span> <span m="3679430">except</span> <span m="3679790">to</span> <span m="3679870">that</span> <span m="3680580">nothing</span> <span m="3680920">was</span> <span m="3681080">easy</span> <span m="3681340">with</span> <span m="3681480">Frank</span> <span m="3681740">Gehry,</span> <span m="3682490">everything</span> <span m="3682870">was</span> <span m="3682970">some</span> <span m="3683290">weird,</span> <span m="3683890">weird</span> <span m="3684190">thing</span> <span m="3684370">going</span> <span m="3684610">on.</span> <span m="3685310">And</span> <span m="3685510">he</span> <span m="3685740">insisted</span> <span m="3686250">on</span> <span m="3686370">using</span> <span m="3686790">l-shaped</span> <span m="3687560">tiles</span> <span m="3688860">for</span> <span m="3688950">the</span> <span m="3689030">courtyard.</span> </p>
<p><span m="3690660">So</span> <span m="3690960">the</span> <span m="3691080">tiles</span> <span m="3691740">that</span> <span m="3691820">we're</span> <span m="3691910">going</span> <span m="3692010">to</span> <span m="3692060">use</span> <span m="3692510">looked</span> <span m="3692710">like</span> <span m="3693600">this.</span> <span m="3694560">So</span> <span m="3694840">it's</span> <span m="3696390">almost</span> <span m="3696730">a</span> <span m="3696790">two</span> <span m="3696980">by</span> <span m="3697170">two,</span> <span m="3697770">except</span> <span m="3698160">you're</span> <span m="3698260">missing</span> <span m="3699260">that</span> <span m="3699460">piece.</span> <span m="3701220">So</span> <span m="3701430">what you</span> <span m="3701530">need</span> <span m="3701710">to</span> <span m="3701780">figure</span> <span m="3702030">out</span> <span m="3702190">how</span> <span m="3702310">to</span> <span m="3702430">do</span> <span m="3702930">is</span> <span m="3703600">tie</span> <span m="3703920">all</span> <span m="3704030">this</span> <span m="3704250">courtyard</span> <span m="3704870">perfectly</span> <span m="3706170">leaving</span> <span m="3707000">one</span> <span m="3707190">spot</span> <span m="3707490">for</span> <span m="3707590">Bill</span> <span m="3708550">using</span> <span m="3708900">tiles</span> <span m="3709530">like</span> <span m="3709760">this,</span> <span m="3710060">these</span> <span m="3710290">l-shaped</span> <span m="3710870">tiles.</span> <span m="3717400">And</span> <span m="3717500">this</span> <span m="3717670">is</span> <span m="3717810">2</span> <span m="3717970">by</span> <span m="3718120">2</span> <span m="3718360">here.</span> <span m="3721460">So</span> <span m="3721590">that's</span> <span m="3721900">the</span> <span m="3722000">task.</span> </p>
<p><span m="3723150">So</span> <span m="3723790">let's</span> <span m="3723970">see</span> <span m="3724070">if</span> <span m="3724180">we</span> <span m="3724270">can</span> <span m="3724420">do</span> <span m="3724610">that</span> <span m="3725740">for</span> <span m="3725870">n</span> <span m="3726010">equals</span> <span m="3726290">2</span> <span m="3726540">here.</span> <span m="3731090">Let's</span> <span m="3731350">see,</span> <span m="3731470">we</span> <span m="3731610">can</span> <span m="3731750">do</span> <span m="3732230">a</span> <span m="3732290">tile</span> <span m="3732640">here.</span> <span m="3735040">We</span> <span m="3735330">can</span> <span m="3735520">do</span> <span m="3735860">a</span> <span m="3735940">tile</span> <span m="3736930">here.</span> <span m="3739100">A</span> <span m="3739150">tile</span> <span m="3739480">here.</span> <span m="3740870">A</span> <span m="3741000">tile</span> <span m="3741380">here.</span> <span m="3742520">And</span> <span m="3742610">a</span> <span m="3742670">tile</span> <span m="3742960">there.</span> <span m="3744730">So</span> <span m="3744900">we</span> <span m="3745010">can.</span> <span m="3745320">In</span> <span m="3745390">this</span> <span m="3745550">case</span> <span m="3746330">we</span> <span m="3746540">can</span> <span m="3747090">tile</span> <span m="3747460">the</span> <span m="3747550">courtyard</span> <span m="3748000">perfectly</span> <span m="3748530">using</span> <span m="3749720">these</span> <span m="3749970">L</span> <span m="3750160">shaped</span> <span m="3750440">tiles,</span> <span m="3751330">leaving</span> <span m="3751670">one</span> <span m="3751990">square</span> <span m="3752440">in</span> <span m="3752500">the</span> <span m="3752600">Center</span> <span m="3753890">for</span> <span m="3753990">the</span> <span m="3754100">statue.</span> </p>
<p><span m="3756850">All right,</span> <span m="3756910">everyone</span> <span m="3757150">understand</span> <span m="3757640">what</span> <span m="3757750">we're</span> <span m="3757820">trying</span> <span m="3758080">to</span> <span m="3758140">do?</span> <span m="3759110">The</span> <span m="3759200">goal?</span> <span m="3760790">Now</span> <span m="3760920">I</span> <span m="3761000">want</span> <span m="3761160">to</span> <span m="3761230">do</span> <span m="3761400">it</span> <span m="3761480">for</span> <span m="3761939">n.</span> <span m="3766070">So</span> <span m="3766240">let's</span> <span m="3767620">start</span> <span m="3768180">proving</span> <span m="3768710">it</span> <span m="3769130">by</span> <span m="3770120">induction,</span> <span m="3771270">even</span> <span m="3771490">though</span> <span m="3771800">we</span> <span m="3771900">don't</span> <span m="3772050">know</span> <span m="3772140">how</span> <span m="3772240">to</span> <span m="3772350">do</span> <span m="3772540">it</span> <span m="3772630">yet.</span> <span m="3772790">Because</span> <span m="3772910">we're</span> <span m="3772990">going</span> <span m="3773070">to</span> <span m="3773130">see</span> <span m="3773280">how</span> <span m="3773420">induction's</span> <span m="3773970">going</span> <span m="3774080">to</span> <span m="3774140">help</span> <span m="3774410">us</span> <span m="3775330">show</span> <span m="3775670">it's</span> <span m="3775790">possible</span> <span m="3776480">and</span> <span m="3776560">maybe</span> <span m="3776760">even</span> <span m="3777630">show</span> <span m="3777820">us</span> <span m="3777950">how</span> <span m="3778140">to</span> <span m="3778250">do it.</span> </p>
<p><span m="3783700">So</span> <span m="3783860">let's</span> <span m="3784050">state</span> <span m="3784270">a</span> <span m="3784330">theorem.</span> <span m="3787970">For</span> <span m="3788200">all</span> <span m="3788390">n</span> <span m="3790070">there</span> <span m="3790260">exists</span> <span m="3790780">away</span> <span m="3793600">to</span> <span m="3793840">tile</span> <span m="3796280">a</span> <span m="3796570">2</span> <span m="3796860">to</span> <span m="3796940">the</span> <span m="3797110">n</span> <span m="3797860">by</span> <span m="3798260">2</span> <span m="3798470">to</span> <span m="3798540">the</span> <span m="3798710">n</span> <span m="3800410">region,</span> <span m="3802420">or</span> <span m="3802570">courtyard,</span> <span m="3805310">with</span> <span m="3806030">a</span> <span m="3807160">center</span> <span m="3807680">square</span> <span m="3808120">missing</span> <span m="3808840">for</span> <span m="3809120">Bill.</span> <span m="3823230">And</span> <span m="3823400">the</span> <span m="3823480">proof</span> <span m="3823780">will</span> <span m="3823860">be</span> <span m="3824000">by</span> <span m="3824220">induction.</span> </p>
<p><span m="3832670">And</span> <span m="3833010">our</span> <span m="3833140">induction</span> <span m="3833570">hypothesis,</span> <span m="3834440">the</span> <span m="3834520">predicate,</span> <span m="3836250">pn</span> <span m="3836830">is</span> <span m="3836970">going</span> <span m="3837210">to</span> <span m="3837280">be</span> <span m="3838080">what</span> <span m="3838210">we're</span> <span m="3838280">trying</span> <span m="3838470">to</span> <span m="3838530">prove.</span> <span m="3841450">So</span> <span m="3841600">this</span> <span m="3842060">is</span> <span m="3842390">the</span> <span m="3842570">induction</span> <span m="3842920">hypothesis.</span> <span m="3845960">Almost</span> <span m="3846370">always</span> <span m="3846880">when</span> <span m="3846980">you</span> <span m="3847070">do</span> <span m="3847200">the</span> <span m="3847290">induction</span> <span m="3847680">you</span> <span m="3847770">want</span> <span m="3847880">to</span> <span m="3847940">start</span> <span m="3848180">out</span> <span m="3848320">with</span> <span m="3848430">that</span> <span m="3848620">as</span> <span m="3848720">your</span> <span m="3848860">hypothesis.</span> <span m="3851380">What's</span> <span m="3851570">the</span> <span m="3851640">next</span> <span m="3851830">step?</span> <span m="3854350">Base</span> <span m="3854680">case.</span> <span m="3855040">Never</span> <span m="3855490">ever</span> <span m="3855780">forget</span> <span m="3856100">the</span> <span m="3856170">base</span> <span m="3856460">case</span> <span m="3858460">or</span> <span m="3858710">you'll</span> <span m="3858830">be</span> <span m="3859340">thinking all</span> <span m="3859590">horses</span> <span m="3860140">are</span> <span m="3860180">the</span> <span m="3860290">same</span> <span m="3860570">color.</span> <span m="3862660">p0.</span> </p>
<p><span m="3865350">Well,</span> <span m="3865970">the</span> <span m="3866070">courtyard</span> <span m="3866800">for</span> <span m="3866920">n</span> <span m="3867080">equals</span> <span m="3867410">0</span> <span m="3868980">is</span> <span m="3869160">just</span> <span m="3869670">1</span> <span m="3870090">square.</span> <span m="3872190">And</span> <span m="3872430">that's</span> <span m="3872640">for</span> <span m="3872740">Bill.</span> <span m="3875150">So</span> <span m="3875350">you're</span> <span m="3875460">done.</span> <span m="3876230">There's</span> <span m="3876460">no</span> <span m="3876700">tiles</span> <span m="3877110">at</span> <span m="3877180">all</span> <span m="3877340">to</span> <span m="3877410">worry</span> <span m="3877640">about.</span> <span m="3878790">That's</span> <span m="3879190">easy.</span> <span m="3880500">So</span> <span m="3880700">that's</span> <span m="3880860">true.</span> <span m="3883070">And</span> <span m="3883290">then</span> <span m="3883480">we</span> <span m="3883570">do</span> <span m="3883720">the</span> <span m="3883910">inductive</span> <span m="3884410">step.</span> </p>
<p><span m="3911270">So</span> <span m="3913280">inductive</span> <span m="3913650">step.</span> <span m="3919650">For</span> <span m="3920290">n</span> <span m="3921100">bigger</span> <span m="3921350">than or</span> <span m="3921450">equal</span> <span m="3921570">to</span> <span m="3921640">0,</span> <span m="3922050">got</span> <span m="3922200">to</span> <span m="3922420">remember</span> <span m="3922640">to</span> <span m="3922710">keep</span> <span m="3922910">track</span> <span m="3923150">of</span> <span m="3923220">that</span> <span m="3923430">now,</span> <span m="3925030">we</span> <span m="3925190">assume</span> <span m="3926950">pn</span> <span m="3929020">to</span> <span m="3929230">verify</span> <span m="3929810">the</span> <span m="3929920">inductive</span> <span m="3930330">hypothesis,</span> <span m="3931240">or to</span> <span m="3931430">prove</span> <span m="3931780">pn</span> <span m="3932070">plus</span> <span m="3932330">1.</span> <span m="3934010">A</span> <span m="3934520">lot</span> <span m="3934730">of</span> <span m="3934800">ways</span> <span m="3935050">to</span> <span m="3935160">write</span> <span m="3935390">this</span> <span m="3935580">down,</span> <span m="3935930">but</span> <span m="3936080">you</span> <span m="3936170">always</span> <span m="3936440">want</span> <span m="3936610">to</span> <span m="3937040">say</span> <span m="3937190">what</span> <span m="3937390">you're</span> <span m="3937500">assuming</span> <span m="3937880">and</span> <span m="3937960">why.</span> </p>
<p><span m="3940650">So</span> <span m="3940910">we</span> <span m="3941010">need</span> <span m="3941220">to</span> <span m="3941290">show</span> <span m="3945050">pn</span> <span m="3945450">plus</span> <span m="3945650">1</span> <span m="3945810">is</span> <span m="3945900">true.</span> <span m="3952940">Well,</span> <span m="3953160">so</span> <span m="3953270">let's</span> <span m="3953510">look</span> <span m="3953710">at</span> <span m="3953770">a</span> <span m="3953950">2</span> <span m="3954130">to</span> <span m="3954190">the</span> <span m="3954360">n</span> <span m="3954670">by--</span> <span m="3955400">2 to</span> <span m="3955530">the</span> <span m="3955690">n</span> <span m="3955780">plus</span> <span m="3956000">1</span> <span m="3956280">by</span> <span m="3956410">2</span> <span m="3956590">to</span> <span m="3956630">the</span> <span m="3956720">n</span> <span m="3956820">plus</span> <span m="3957020">1</span> <span m="3957310">courtyard.</span> <span m="3970600">So</span> <span m="3970790">let's</span> <span m="3970990">draw</span> <span m="3971200">it out</span> <span m="3971380">here.</span> <span m="3974450">2 to</span> <span m="3974660">the</span> <span m="3974780">n</span> <span m="3974910">plus</span> <span m="3975170">1</span> <span m="3975770">by</span> <span m="3975940">2</span> <span m="3976100">to</span> <span m="3976160">the</span> <span m="3976260">n</span> <span m="3976390">plus</span> <span m="3976660">1.</span> </p>
<p><span m="3980210">What</span> <span m="3980420">are</span> <span m="3980460">we</span> <span m="3980590">going</span> <span m="3980710">to</span> <span m="3980850">do</span> <span m="3981770">to</span> <span m="3981890">use</span> <span m="3982140">our</span> <span m="3982250">inductive</span> <span m="3982600">hypothesis?</span> <span m="3983400">Yeah?</span> </p>
<p><span m="3984380">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="3999580">PROFESSOR: For</span> <span m="3999820">the--</span> <span m="4000610">yeah.</span> <span m="4001170">So</span> <span m="4001770">you're</span> <span m="4001910">on</span> <span m="4002010">a</span> <span m="4002060">good</span> <span m="4002270">track</span> <span m="4002650">there,</span> <span m="4002820">for</span> <span m="4002940">sure.</span> <span m="4003920">But</span> <span m="4004100">I've</span> <span m="4004180">got</span> <span m="4004270">to</span> <span m="4004310">apply--</span> <span m="4004890">I'm</span> <span m="4004960">not</span> <span m="4005100">going</span> <span m="4005250">from</span> <span m="4005420">1</span> <span m="4005640">to</span> <span m="4005730">2,</span> <span m="4006720">I</span> <span m="4006800">want</span> <span m="4007020">to</span> <span m="4007080">get--</span> <span m="4007570">I</span> <span m="4007630">want</span> <span m="4007790">to</span> <span m="4007830">use</span> <span m="4008030">pn</span> <span m="4008430">here.</span> <span m="4009590">So</span> <span m="4009700">I've</span> <span m="4009760">got</span> <span m="4009960">a</span> <span m="4010030">2 to</span> <span m="4010250">the</span> <span m="4010370">n plus</span> <span m="4010600">1 by</span> <span m="4010730">2</span> <span m="4010880">to</span> <span m="4011000">the</span> <span m="4011110">n</span> <span m="4011180">plus</span> <span m="4011360">1</span> <span m="4011500">courtyard.</span> <span m="4011960">How</span> <span m="4012100">do I</span> <span m="4012240">use</span> <span m="4012490">pn?</span> <span m="4014610">2</span> <span m="4014740">to the</span> <span m="4014950">n</span> <span m="4015060">by</span> <span m="4015160">2 to</span> <span m="4015540">the</span> <span m="4015920">n courtyard.</span> <span m="4016310">Yeah?</span> </p>
<p><span m="4017158">AUDIENCE: Oh, never mind.</span> <span m="4018006">I don't think</span> <span m="4018430">it works.</span> <span m="4018870">I was about</span> <span m="4019362">to say</span> <span m="4019854">that</span> <span m="4020346">would be</span> <span m="4020838">as if 2 to the</span> <span m="4021330">n would</span> <span m="4021822">divide</span> <span m="4022314">that into</span> <span m="4022806">four blocks.</span> </p>
<p><span m="4024282">PROFESSOR: Good idea,</span> <span m="4024774">yeah.</span> <span m="4025770">Let's</span> <span m="4026040">divide</span> <span m="4026330">our</span> <span m="4026440">courtyard</span> <span m="4026720">into</span> <span m="4026990">four</span> <span m="4027260">blocks.</span> <span m="4027830">That's</span> <span m="4028030">a</span> <span m="4028070">great</span> <span m="4028310">idea.</span> <span m="4029410">And</span> <span m="4029640">now</span> <span m="4029800">each</span> <span m="4030030">of</span> <span m="4030090">these</span> <span m="4030330">is</span> <span m="4030440">2</span> <span m="4030610">to</span> <span m="4030670">the</span> <span m="4030810">n</span> <span m="4030960">by</span> <span m="4031090">2</span> <span m="4031270">to</span> <span m="4031330">the</span> <span m="4031500">n.</span> <span m="4033550">Right?</span> <span m="4035210">And</span> <span m="4035430">I</span> <span m="4035500">can</span> <span m="4035630">apply</span> <span m="4035970">the</span> <span m="4036090">inductive</span> <span m="4036500">hypothesis</span> <span m="4037370">there.</span> </p>
<p><span m="4039452">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="4045480">PROFESSOR: Yeah.</span> <span m="4046490">Hm.</span> <span m="4048620">Yeah,</span> <span m="4048800">something--</span> <span m="4049250">yeah, it</span> <span m="4049530">doesn't</span> <span m="4049850">quite</span> <span m="4050110">work</span> <span m="4050560">because</span> <span m="4051620">Bill</span> <span m="4051850">wants</span> <span m="4052100">to</span> <span m="4052160">be</span> <span m="4052290">here.</span> <span m="4054250">Got a</span> <span m="4054400">little</span> <span m="4054530">square</span> <span m="4054820">for</span> <span m="4054920">Bill</span> <span m="4055210">there.</span> <span m="4056280">But</span> <span m="4056430">I</span> <span m="4056510">can't</span> <span m="4057800">use</span> <span m="4058020">my</span> <span m="4058190">inductive</span> <span m="4058550">hypothesis</span> <span m="4059240">to</span> <span m="4059370">tile</span> <span m="4059780">this,</span> <span m="4061760">because</span> <span m="4061930">there's</span> <span m="4062080">no</span> <span m="4062260">square</span> <span m="4062620">missing.</span> <span m="4062980">In</span> <span m="4063050">fact,</span> <span m="4064025">even if</span> <span m="4064440">there</span> <span m="4064600">wasn't</span> <span m="4064810">a</span> <span m="4064910">square</span> <span m="4065220">missing,</span> <span m="4065500">I'm</span> <span m="4065610">in</span> <span m="4065680">trouble.</span> </p>
<p><span m="4066190">If</span> <span m="4066330">I've</span> <span m="4066450">got--</span> <span m="4067490">say</span> <span m="4067700">this</span> <span m="4067930">is</span> <span m="4068040">size</span> <span m="4068640">4</span> <span m="4068930">by</span> <span m="4069060">4</span> <span m="4069340">here.</span> <span m="4070480">Can I</span> <span m="4070820">tile</span> <span m="4071160">a</span> <span m="4071220">4</span> <span m="4071430">by</span> <span m="4071550">4</span> <span m="4071810">region</span> <span m="4072140">with</span> <span m="4072310">L</span> <span m="4072470">shaped</span> <span m="4072700">tiles?</span> <span m="4075390">No, they're</span> <span m="4075880">size</span> <span m="4076220">3.</span> <span m="4076475">3</span> <span m="4076730">doesn't</span> <span m="4076830">divide</span> <span m="4076900">into</span> <span m="4077010">16.</span> <span m="4077420">Yeah?</span> </p>
<p><span m="4079282">AUDIENCE: [INAUDIBLE]</span> <span m="4081178">There's one tile</span> <span m="4081652">missing from</span> <span m="4082126">each of those</span> <span m="4082600">blocks</span> <span m="4083074">at the corner</span> <span m="4083548">that</span> <span m="4084022">[INAUDIBLE].</span> </p>
<p><span m="4086300">PROFESSOR: There's</span> <span m="4086610">a</span> <span m="4086650">great</span> <span m="4087000">idea.</span> <span m="4087400">All</span> <span m="4087640">right,</span> <span m="4087870">we're</span> <span m="4088190">making</span> <span m="4088470">progress</span> <span m="4089050">now.</span> <span m="4090030">Take</span> <span m="4090290">a</span> <span m="4090360">corner</span> <span m="4090750">out</span> <span m="4090870">of</span> <span m="4090970">each</span> <span m="4091170">of</span> <span m="4091250">them.</span> <span m="4092640">Put</span> <span m="4092830">my</span> <span m="4092990">l-shaped</span> <span m="4093550">tile</span> <span m="4094000">here.</span> <span m="4096529">Now</span> <span m="4097670">I</span> <span m="4097790">can</span> <span m="4097939">use the</span> <span m="4098130">inductive</span> <span m="4098550">hypothesis</span> <span m="4099200">to</span> <span m="4099319">tile</span> <span m="4100310">each</span> <span m="4100560">one</span> <span m="4100800">of</span> <span m="4100890">these.</span> <span m="4104750">Yeah?</span> <span m="4106643">Yeah?</span> </p>
<p><span m="4108137">AUDIENCE: Well,</span> <span m="4108635">why can't</span> <span m="4109133">you put</span> <span m="4109631">a 4</span> <span m="4110129">in the</span> <span m="4110627">center</span> <span m="4111623">and then you</span> <span m="4112121">have a</span> <span m="4112619">bunch of</span> <span m="4113117">2s</span> <span m="4113615">on the side?</span> </p>
<p><span m="4114113">PROFESSOR: A</span> <span m="4114620">4 in the</span> <span m="4114990">center--</span> </p>
<p><span m="4115689">AUDIENCE: Like a</span> <span m="4116119">2 by</span> <span m="4116549">2</span> <span m="4116979">in the center.</span> </p>
<p><span m="4118269">PROFESSOR: I got that.</span> <span m="4118700">I got</span> <span m="4119015">Bill and</span> <span m="4119330">the tile.</span> </p>
<p><span m="4120790">AUDIENCE: No,</span> <span m="4121272">but like</span> <span m="4121754">make it</span> <span m="4122236">bigger.</span> <span m="4124147">Not just</span> <span m="4124634">like</span> <span m="4125121">4 single</span> <span m="4125608">tiles,</span> <span m="4126095">but like--</span> <span m="4127069">so you have</span> <span m="4127556">something</span> <span m="4128043">like that over</span> <span m="4128530">there,</span> <span m="4129017">right?</span> <span m="4129991">Put that in</span> <span m="4130478">the center.</span> </p>
<p><span m="4130965">PROFESSOR: Put that in</span> <span m="4131452">the center,</span> <span m="4131939">OK.</span> </p>
<p><span m="4132913">AUDIENCE: And the rest</span> <span m="4133400">of them</span> <span m="4133887">will have</span> <span m="4134374">like</span> <span m="4134861">[INAUDIBLE].</span> </p>
<p><span m="4136529">PROFESSOR: Well,</span> <span m="4136680">the</span> <span m="4136800">rest</span> <span m="4137080">of</span> <span m="4137140">them</span> <span m="4137330">aren't</span> <span m="4137910">2</span> <span m="4138210">by</span> <span m="4138490">2,</span> <span m="4138720">because</span> <span m="4138890">this</span> <span m="4139060">is</span> <span m="4139210">n.</span> <span m="4140399">I've</span> <span m="4140490">got</span> <span m="4140670">to</span> <span m="4140710">use</span> <span m="4140970">pn</span> <span m="4141350">here.</span> <span m="4141600">And</span> <span m="4141689">pn</span> <span m="4142109">says</span> <span m="4142529">that</span> <span m="4142660">I</span> <span m="4142729">can--</span> <span m="4143540">in</span> <span m="4143689">a</span> <span m="4143819">region</span> <span m="4144260">2 to</span> <span m="4144390">the</span> <span m="4144560">n</span> <span m="4144670">by</span> <span m="4144779">2 to</span> <span m="4145029">the</span> <span m="4145189">n</span> <span m="4145380">with</span> <span m="4145540">a</span> <span m="4145590">square</span> <span m="4145950">out</span> <span m="4146040">of</span> <span m="4146100">the</span> <span m="4146189">center,</span> <span m="4146670">I</span> <span m="4146729">can</span> <span m="4147039">tile</span> <span m="4147350">it.</span> <span m="4149970">Am</span> <span m="4150040">I</span> <span m="4150279">good</span> <span m="4150439">here</span> <span m="4150630">so</span> <span m="4150810">far?</span> <span m="4151310">I</span> <span m="4151380">claimed</span> <span m="4151620">I</span> <span m="4151760">was</span> <span m="4151840">sort</span> <span m="4152020">of</span> <span m="4152120">done.</span> <span m="4153020">Yeah?</span> </p>
<p><span m="4154271">AUDIENCE: No,</span> <span m="4155255">because</span> <span m="4155747">p of</span> <span m="4156239">n</span> <span m="4156731">tells us--</span> <span m="4157715">well,</span> <span m="4158207">assume</span> <span m="4158699">p of n</span> <span m="4159191">tells us that</span> <span m="4159683">you can set</span> <span m="4160175">up a</span> <span m="4160667">2 to the n</span> <span m="4161159">by 2 to the n</span> <span m="4161651">courtyard with</span> <span m="4162143">a centerpiece</span> <span m="4162635">of--</span> </p>
<p><span m="4163140">PROFESSOR: Yeah.</span> </p>
<p><span m="4164200">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="4165779">PROFESSOR: Yeah.</span> <span m="4166350">And</span> <span m="4166430">what's</span> <span m="4166689">the</span> <span m="4166760">problem</span> <span m="4167319">for</span> <span m="4167420">this</span> <span m="4167760">2 to the</span> <span m="4167920">n</span> <span m="4168090">by</span> <span m="4168189">2 to the</span> <span m="4168444">n</span> <span m="4168700">region?</span> <span m="4171029">Bill's not in the</span> <span m="4171290">center.</span> <span m="4171620">He</span> <span m="4171750">wants</span> <span m="4171960">to</span> <span m="4172020">be</span> <span m="4172090">in</span> <span m="4172149">the</span> <span m="4172229">center.</span> <span m="4173170">We put</span> <span m="4173580">Bill</span> <span m="4173880">in the</span> <span m="4173960">corner.</span> <span m="4175529">So</span> <span m="4175660">you</span> <span m="4175729">can't</span> <span m="4176010">use the</span> <span m="4176300">inductive</span> <span m="4176720">hypothesis</span> <span m="4177359">here.</span> <span m="4180130">So</span> <span m="4180270">I</span> <span m="4180760">went</span> <span m="4180810">a</span> <span m="4180859">little</span> <span m="4181000">too</span> <span m="4181130">fast,</span> <span m="4181510">here.</span> <span m="4181899">This</span> <span m="4182080">is</span> <span m="4182189">not</span> <span m="4182380">a proof</span> <span m="4182689">so</span> <span m="4182819">far.</span> <span m="4184450">I've</span> <span m="4184540">got</span> <span m="4184670">a</span> <span m="4184710">problem.</span> <span m="4185970">Yeah?</span> </p>
<p><span m="4187529">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="4195260">PROFESSOR: Great.</span> <span m="4195710">OK,</span> <span m="4196050">let's</span> <span m="4196560">then</span> <span m="4197110">change</span> <span m="4197690">the</span> <span m="4197930">inductive</span> <span m="4198430">hypothesis.</span> <span m="4200060">Good.</span> <span m="4200750">Let's</span> <span m="4201150">say</span> <span m="4201800">there</span> <span m="4201910">exists</span> <span m="4202240">a</span> <span m="4202300">way</span> <span m="4202460">to</span> <span m="4202570">tile</span> <span m="4202920">a 2 to</span> <span m="4203140">the</span> <span m="4203310">n</span> <span m="4203460">by</span> <span m="4203570">2 to</span> <span m="4203670">the n</span> <span m="4203920">region</span> <span m="4204810">with</span> <span m="4205090">a</span> <span m="4205210">corner</span> <span m="4205710">square</span> <span m="4206040">missing</span> <span m="4210530">for Bill.</span> <span m="4214050">All</span> <span m="4214260">right,</span> <span m="4218030">but</span> <span m="4218200">now I</span> <span m="4218380">got</span> <span m="4218510">to</span> <span m="4218560">put--</span> <span m="4219400">oh,</span> <span m="4219550">but</span> <span m="4219650">I</span> <span m="4219700">can</span> <span m="4219850">make</span> <span m="4220020">this</span> <span m="4220180">work.</span> <span m="4220500">Yeah.</span> <span m="4220770">Yeah.</span> <span m="4220930">We</span> <span m="4221030">can</span> <span m="4221140">make</span> <span m="4221350">something--</span> <span m="4221470">we</span> <span m="4221590">can</span> <span m="4221760">prove</span> <span m="4222000">this</span> <span m="4222240">now.</span> </p>
<p><span m="4222760">The</span> <span m="4222860">inductive</span> <span m="4223230">step is</span> <span m="4223510">going</span> <span m="4223610">to</span> <span m="4223680">work</span> <span m="4223970">because</span> <span m="4225290">I'll</span> <span m="4225450">put</span> <span m="4225660">Bill--</span> <span m="4226050">say he's</span> <span m="4226320">in</span> <span m="4226590">one</span> <span m="4226800">of</span> <span m="4226840">the</span> <span m="4226910">four</span> <span m="4227170">regions.</span> <span m="4228320">Then</span> <span m="4228550">use</span> <span m="4228760">the</span> <span m="4228920">l</span> <span m="4229230">down</span> <span m="4229520">here.</span> <span m="4231510">Put</span> <span m="4231790">the l</span> <span m="4231920">here.</span> <span m="4232850">And now</span> <span m="4233130">I've</span> <span m="4233240">got</span> <span m="4233360">a</span> <span m="4233410">corner</span> <span m="4233870">out</span> <span m="4234040">of</span> <span m="4234130">each</span> <span m="4234320">one.</span> <span m="4234900">And</span> <span m="4235150">now</span> <span m="4235320">I'm</span> <span m="4235460">done</span> <span m="4235680">by</span> <span m="4235820">induction.</span> <span m="4237810">I</span> <span m="4237970">have</span> <span m="4238225">proved</span> <span m="4238480">there's</span> <span m="4238640">a</span> <span m="4238720">way</span> <span m="4238910">to</span> <span m="4239030">tile</span> <span m="4240460">any</span> <span m="4240720">2 to</span> <span m="4240940">the</span> <span m="4241050">n</span> <span m="4241315">by 2 to</span> <span m="4241580">the n</span> <span m="4241700">region</span> <span m="4242930">with</span> <span m="4243110">a</span> <span m="4243180">corner</span> <span m="4243590">square</span> <span m="4243910">missing.</span> <span m="4246050">I</span> <span m="4246150">think</span> <span m="4246310">that</span> <span m="4246480">proof</span> <span m="4246680">works,</span> <span m="4247442">Yeah?</span> </p>
<p><span m="4248426">AUDIENCE: [INAUDIBLE]</span> <span m="4253346">with Bill in the</span> <span m="4253838">middle,</span> <span m="4254330">because</span> <span m="4254822">to</span> <span m="4255314">prove</span> <span m="4255806">that you</span> <span m="4256298">can put Bill</span> <span m="4256790">in the corner</span> <span m="4257282">in a</span> <span m="4257774">2</span> <span m="4258266">by 2 case,</span> <span m="4258758">you can</span> <span m="4259250">just</span> <span m="4259742">blow that</span> <span m="4260234">square off.</span> <span m="4261218">[INAUDIBLE]</span> <span m="4262694">just becomes</span> <span m="4263186">just another</span> <span m="4263678">square,</span> <span m="4264170">which is</span> <span m="4264662">4 by</span> <span m="4265154">4</span> <span m="4265646">and then</span> <span m="4266138">you can</span> <span m="4266630">put him in</span> <span m="4267122">the middle.</span> </p>
<p><span m="4268140">PROFESSOR: That is true.</span> <span m="4268770">So you've jumped</span> <span m="4269050">ahead.</span> <span m="4270140">We</span> <span m="4270390">have</span> <span m="4270680">successfully</span> <span m="4271310">now</span> <span m="4271390">proved</span> <span m="4272300">this</span> <span m="4272750">is</span> <span m="4272880">true</span> <span m="4273200">for</span> <span m="4273350">p</span> <span m="4273560">of</span> <span m="4273650">n.</span> <span m="4275050">But</span> <span m="4275190">Bill</span> <span m="4275600">didn't</span> <span m="4275630">want</span> <span m="4275870">to</span> <span m="4275920">be</span> <span m="4276040">in</span> <span m="4276130">the</span> <span m="4276230">corner.</span> <span m="4276550">We're</span> <span m="4276700">trying</span> <span m="4276960">to</span> <span m="4277040">prove</span> <span m="4277380">there's</span> <span m="4277520">a</span> <span m="4277580">way</span> <span m="4277720">to</span> <span m="4277820">do</span> <span m="4277940">it with</span> <span m="4278110">Bill</span> <span m="4278320">in</span> <span m="4278380">the</span> <span m="4278460">center,</span> <span m="4279810">which</span> <span m="4279960">is</span> <span m="4280060">not</span> <span m="4280340">what we</span> <span m="4280480">proved.</span> <span m="4280760">And</span> <span m="4280810">you've</span> <span m="4280990">come</span> <span m="4281190">up</span> <span m="4281290">with</span> <span m="4281400">a</span> <span m="4281450">way--</span> <span m="4282200">yeah,</span> <span m="4282370">that</span> <span m="4282500">might</span> <span m="4282650">be</span> <span m="4282750">doable.</span> </p>
<p><span m="4283120">First</span> <span m="4283530">prove</span> <span m="4283730">you</span> <span m="4283810">can</span> <span m="4283960">put</span> <span m="4284030">Bill</span> <span m="4284290">in</span> <span m="4284340">the</span> <span m="4284440">corner.</span> <span m="4285690">And</span> <span m="4287600">do</span> <span m="4287800">it</span> <span m="4287860">in</span> <span m="4288160">2 to</span> <span m="4288380">the</span> <span m="4288510">n</span> <span m="4288630">by</span> <span m="4288740">2 to</span> <span m="4288970">the</span> <span m="4289140">n's.</span> <span m="4291420">I</span> <span m="4291550">don't</span> <span m="4291750">know</span> <span m="4291960">about</span> <span m="4292220">this,</span> <span m="4292440">actually.</span> <span m="4294300">I</span> <span m="4294350">got</span> <span m="4294530">Bill</span> <span m="4294750">in</span> <span m="4294800">the</span> <span m="4294900">corner.</span> <span m="4295890">There's</span> <span m="4296040">some</span> <span m="4296290">way</span> <span m="4296420">to</span> <span m="4296530">do</span> <span m="4296780">it.</span> <span m="4298010">But</span> <span m="4298140">it</span> <span m="4298200">might</span> <span m="4298450">have</span> <span m="4298600">involved</span> <span m="4299030">doing</span> <span m="4299300">this.</span> <span m="4299630">And</span> <span m="4299710">now</span> <span m="4299840">I</span> <span m="4299910">can't</span> <span m="4300250">rotate</span> <span m="4300900">that.</span> <span m="4301910">I</span> <span m="4302030">don't</span> <span m="4302170">think</span> <span m="4302300">we</span> <span m="4302420">have--</span> <span m="4302650">I don't</span> <span m="4302760">think that</span> <span m="4303020">proof</span> <span m="4303360">necessarily</span> <span m="4303900">works.</span> </p>
<p><span m="4305115">AUDIENCE: [INAUDIBLE]</span> <span m="4313035">prove</span> <span m="4313530">that he could be</span> <span m="4314025">here.</span> <span m="4314520">But likewise,</span> <span m="4315015">we could</span> <span m="4315510">prove</span> <span m="4316005">that</span> <span m="4316500">if the</span> <span m="4316995">block just</span> <span m="4317490">rotates</span> <span m="4317985">you   can</span> <span m="4318480">[INAUDIBLE].</span> </p>
<p><span m="4319470">PROFESSOR: Yes,</span> <span m="4319965">I agree with you</span> <span m="4320470">and I liked it</span> <span m="4321170">when</span> <span m="4321290">I</span> <span m="4321360">first</span> <span m="4321660">heard it.</span> <span m="4321960">And you</span> <span m="4322100">might</span> <span m="4322310">still</span> <span m="4322550">convince</span> <span m="4322980">me.</span> <span m="4323860">But</span> <span m="4324550">all</span> <span m="4324860">we</span> <span m="4325000">proved</span> <span m="4326590">is</span> <span m="4327070">this,</span> <span m="4327550">there's</span> <span m="4327750">a 2 to</span> <span m="4327980">the</span> <span m="4328100">n</span> <span m="4328230">by 2</span> <span m="4328310">to</span> <span m="4328540">the</span> <span m="4328670">n--</span> <span m="4329370">any</span> <span m="4329520">2 to the</span> <span m="4329660">n by</span> <span m="4329730">2 to the</span> <span m="4329840">n</span> <span m="4330390">region,</span> <span m="4330670">we</span> <span m="4330740">could</span> <span m="4330870">tile</span> <span m="4331270">it with</span> <span m="4331400">a</span> <span m="4331460">corner,</span> <span m="4331940">Bill in</span> <span m="4332230">a</span> <span m="4332280">corner,</span> <span m="4332910">any</span> <span m="4333050">corner</span> <span m="4333310">square</span> <span m="4333510">missing.</span> </p>
<p><span m="4335480">Good,</span> <span m="4336010">so</span> <span m="4336410">now</span> <span m="4336600">you</span> <span m="4336750">want</span> <span m="4336890">to</span> <span m="4336960">say,</span> <span m="4337200">I</span> <span m="4337310">can</span> <span m="4337480">get</span> <span m="4337680">Bill</span> <span m="4337880">in</span> <span m="4337940">the</span> <span m="4338010">middle.</span> <span m="4338910">And</span> <span m="4339100">what</span> <span m="4339240">you</span> <span m="4339330">want</span> <span m="4339560">to</span> <span m="4339630">say</span> <span m="4339900">is</span> <span m="4340040">OK,</span> <span m="4340400">I</span> <span m="4340510">tile</span> <span m="4340910">this</span> <span m="4341140">region</span> <span m="4342330">with</span> <span m="4343000">Bill</span> <span m="4343380">here,</span> <span m="4344090">in</span> <span m="4344190">a</span> <span m="4344270">2</span> <span m="4344460">to</span> <span m="4344530">the</span> <span m="4344680">n</span> <span m="4344880">by--</span> <span m="4350540">maybe</span> <span m="4350780">you're</span> <span m="4350840">right.</span> <span m="4351090">So</span> <span m="4351180">then</span> <span m="4351360">I</span> <span m="4351410">would</span> <span m="4351530">apply</span> <span m="4352580">the</span> <span m="4352700">theorem</span> <span m="4353200">here,</span> <span m="4353650">with</span> <span m="4353780">Bill</span> <span m="4354000">here.</span> <span m="4355150">Oh,</span> <span m="4355290">Bill</span> <span m="4355480">here.</span> <span m="4355940">No,</span> <span m="4356020">I</span> <span m="4356150">think</span> <span m="4356320">I</span> <span m="4356390">like</span> <span m="4356620">it</span> <span m="4356720">now.</span> </p>
<p><span m="4357860">And</span> <span m="4357980">then</span> <span m="4358120">I</span> <span m="4358190">would</span> <span m="4358360">take</span> <span m="4358570">these</span> <span m="4358810">out.</span> </p>
<p><span m="4359546">AUDIENCE: [INAUDIBLE]</span> <span m="4363274">just takes up</span> <span m="4363740">up three</span> <span m="4364130">blocks.</span> <span m="4365044">And</span> <span m="4365958">that big</span> <span m="4366415">square</span> <span m="4366872">is</span> <span m="4367329">just</span> <span m="4367790">like</span> <span m="4368920">a zoomed</span> <span m="4369300">out version</span> <span m="4369780">of a</span> <span m="4370260">little</span> <span m="4370740">square.</span> <span m="4372660">And since you know</span> <span m="4373140">that</span> <span m="4373620">little</span> <span m="4374100">square</span> <span m="4374580">fits in</span> <span m="4375060">a</span> <span m="4375550">2 by</span> <span m="4375720">2--</span> <span m="4375980">like</span> <span m="4376240">for example</span> <span m="4377668">in the</span> <span m="4380700">4 by 4</span> <span m="4382760">case.</span> <span m="4384240">We proved</span> <span m="4384740">the 2 by 2</span> <span m="4385800">case.</span> <span m="4386470">In the 4</span> <span m="4386670">by 4</span> <span m="4386840">case</span> <span m="4388886">you</span> <span m="4389368">have</span> <span m="4389850">a</span> <span m="4390430">big</span> <span m="4390870">size.</span> <span m="4391235">It's</span> <span m="4391600">just</span> <span m="4391900">like</span> <span m="4392320">a--</span> <span m="4393540">you have four squares,</span> <span m="4394010">right?</span> </p>
<p><span m="4394770">And each</span> <span m="4395231">of those</span> <span m="4395692">four</span> <span m="4396153">squares</span> <span m="4396614">has</span> <span m="4397075">[INAUDIBLE].</span> <span m="4398919">So you take</span> <span m="4399850">that</span> <span m="4401390">top</span> <span m="4401790">right</span> <span m="4402190">square</span> <span m="4402940">and</span> <span m="4403260">you put</span> <span m="4403640">Bill</span> <span m="4403965">within</span> <span m="4404290">the</span> <span m="4404718">smaller</span> <span m="4406430">square</span> <span m="4407560">exactly</span> <span m="4407900">where you</span> <span m="4408470">want</span> <span m="4408840">him to</span> <span m="4408940">be.</span> <span m="4409430">And then</span> <span m="4409800">you fill</span> <span m="4410170">those</span> <span m="4410750">other--</span> <span m="4411380">all those</span> <span m="4411850">empty</span> <span m="4412360">spaces.</span> </p>
<p><span m="4413410">PROFESSOR: I</span> <span m="4413700">agree.</span> <span m="4414390">You</span> <span m="4414550">could make</span> <span m="4414710">a</span> <span m="4414970">proof.</span> <span m="4415300">So</span> <span m="4415440">in</span> <span m="4415500">this</span> <span m="4415710">case</span> <span m="4415960">you'd</span> <span m="4416070">make</span> <span m="4416310">a</span> <span m="4416360">Lemma</span> <span m="4417860">that</span> <span m="4418000">uses</span> <span m="4418330">induction</span> <span m="4419580">that</span> <span m="4419720">says</span> <span m="4420170">you</span> <span m="4420290">can</span> <span m="4420470">do it</span> <span m="4420770">with</span> <span m="4420800">Bill</span> <span m="4421050">in</span> <span m="4421090">the</span> <span m="4421190">corner.</span> <span m="4422350">And</span> <span m="4422460">then</span> <span m="4422580">as</span> <span m="4422690">a</span> <span m="4422750">corollary</span> <span m="4423400">or a</span> <span m="4423570">theorem</span> <span m="4424080">you'd</span> <span m="4424260">take</span> <span m="4424530">that</span> <span m="4425840">and</span> <span m="4425990">apply</span> <span m="4426390">it</span> <span m="4426520">to</span> <span m="4426620">four</span> <span m="4427750">sub-squares.</span> <span m="4428980">Put</span> <span m="4429210">Bill</span> <span m="4429570">in</span> <span m="4429660">here.</span> <span m="4430970">Take</span> <span m="4431190">these</span> <span m="4431430">out.</span> <span m="4432030">And</span> <span m="4432160">then</span> <span m="4432230">now you'd</span> <span m="4432610">have</span> <span m="4432730">your</span> <span m="4432870">result</span> <span m="4433250">of Bill</span> <span m="4433500">in the</span> <span m="4433560">center.</span> </p>
<p><span m="4433900">So that is</span> <span m="4434080">is</span> <span m="4434210">a</span> <span m="4434270">way</span> <span m="4434400">to</span> <span m="4434510">do</span> <span m="4434720">it.</span> <span m="4435340">It</span> <span m="4435520">ends</span> <span m="4435710">up</span> <span m="4435800">being</span> <span m="4435970">more</span> <span m="4436210">complicated.</span> <span m="4436860">There</span> <span m="4437010">is</span> <span m="4437120">a</span> <span m="4437200">simpler</span> <span m="4437720">approach.</span> <span m="4438300">But</span> <span m="4438420">that</span> <span m="4438580">is</span> <span m="4438830">a</span> <span m="4439130">way</span> <span m="4439260">that</span> <span m="4439370">works.</span> <span m="4439870">It is</span> <span m="4440080">a</span> <span m="4440140">natural</span> <span m="4440470">thing</span> <span m="4440620">you</span> <span m="4440740">would</span> <span m="4440970">do</span> <span m="4441780">if</span> <span m="4441890">you</span> <span m="4441970">had</span> <span m="4442110">this</span> <span m="4442270">on</span> <span m="4442370">homework.</span> <span m="4443490">Is</span> <span m="4443560">you'd</span> <span m="4443670">think</span> <span m="4443890">of</span> <span m="4443970">a</span> <span m="4444040">different</span> <span m="4444520">thing</span> <span m="4444700">to</span> <span m="4444780">prove</span> <span m="4444980">by</span> <span m="4445130">induction,</span> <span m="4445980">then</span> <span m="4446130">use</span> <span m="4446330">that</span> <span m="4446530">as</span> <span m="4446640">a</span> <span m="4446700">Lemma</span> <span m="4447300">to</span> <span m="4447370">get</span> <span m="4447560">you</span> <span m="4447690">where</span> <span m="4447810">you</span> <span m="4447910">wanted</span> <span m="4448110">to</span> <span m="4448150">go.</span> </p>
<p><span m="4449540">There's</span> <span m="4449730">another</span> <span m="4450000">way</span> <span m="4450140">to</span> <span m="4450250">do</span> <span m="4450410">it</span> <span m="4450520">without</span> <span m="4451300">having</span> <span m="4451590">that</span> <span m="4452050">first</span> <span m="4452360">step.</span> <span m="4454290">And</span> <span m="4454380">that's--</span> <span m="4454700">yeah?</span> </p>
<p><span m="4456870">AUDIENCE: That courtyard</span> <span m="4457840">where</span> <span m="4458810">n equals</span> <span m="4459295">2.</span> <span m="4460750">Divide</span> <span m="4461235">each</span> <span m="4461720">cell</span> <span m="4462205">into</span> <span m="4462690">a</span> <span m="4463175">2</span> <span m="4463660">by</span> <span m="4464145">2--</span> </p>
<p><span m="4468530">PROFESSOR: Yeah.</span> </p>
<p><span m="4468820">AUDIENCE: Something.</span> </p>
<p><span m="4471180">PROFESSOR: Well,</span> <span m="4471640">yeah.</span> <span m="4473780">You</span> <span m="4473870">want</span> <span m="4473970">to</span> <span m="4474030">do</span> <span m="4474160">it</span> <span m="4474573">bottoms</span> <span m="4474986">up.</span> <span m="4475400">You</span> <span m="4475570">want</span> <span m="4475750">to take</span> <span m="4475870">it and</span> <span m="4476940">make</span> <span m="4477120">that</span> <span m="4477350">your</span> <span m="4478240">inductive</span> <span m="4478630">step,</span> <span m="4479280">2</span> <span m="4479450">by</span> <span m="4479600">2's</span> <span m="4479850">inside.</span> <span m="4480310">I</span> <span m="4480330">haven't</span> <span m="4480660">thought</span> <span m="4480890">about</span> <span m="4481090">that</span> <span m="4481230">approach.</span> </p>
<p><span m="4482272">AUDIENCE: [INAUDIBLE]</span> <span m="4494572">So if you</span> <span m="4495064">could do</span> <span m="4495556">that then</span> <span m="4496048">you're</span> <span m="4496540">all set</span> <span m="4497032">except</span> <span m="4497524">for the</span> <span m="4498016">2 by</span> <span m="4498508">2</span> <span m="4500968">where</span> <span m="4501460">Bill</span> <span m="4501952">was.</span> </p>
<p><span m="4504440">PROFESSOR: I'm</span> <span m="4504740">worried</span> <span m="4505070">about</span> <span m="4505400">a lot of</span> <span m="4505740">2 by 2's</span> <span m="4505920">with</span> <span m="4506670">a corner</span> <span m="4507083">missing</span> <span m="4507910">if</span> <span m="4508100">I</span> <span m="4508150">do</span> <span m="4508310">it</span> <span m="4508420">that way.</span> <span m="4508883">I'll think</span> <span m="4509346">about that.</span> <span m="4511620">There</span> <span m="4511790">is--</span> <span m="4512180">let</span> <span m="4512290">me</span> <span m="4512420">get</span> <span m="4512590">to</span> <span m="4512810">another</span> <span m="4515020">approach</span> <span m="4515470">here.</span> <span m="4517510">We</span> <span m="4517640">couldn't</span> <span m="4517990">make</span> <span m="4518160">it</span> <span m="4518220">work</span> <span m="4518420">with</span> <span m="4518520">the</span> <span m="4518610">center.</span> <span m="4519080">We</span> <span m="4519200">could</span> <span m="4519450">make</span> <span m="4519630">it</span> <span m="4519720">work</span> <span m="4519910">with</span> <span m="4520030">a</span> <span m="4520080">corner.</span> <span m="4520460">But</span> <span m="4520590">then</span> <span m="4520710">we</span> <span m="4520770">had</span> <span m="4520840">to</span> <span m="4520900">do</span> <span m="4521020">more</span> <span m="4521210">work</span> <span m="4521430">after.</span> <span m="4522960">One</span> <span m="4523490">general</span> <span m="4523940">technique</span> <span m="4524330">to</span> <span m="4524410">use</span> <span m="4524900">with</span> <span m="4525030">induction,</span> <span m="4525650">when</span> <span m="4525750">you're</span> <span m="4525850">having</span> <span m="4526110">struggling,</span> <span m="4526690">what's</span> <span m="4526990">the</span> <span m="4527090">induction</span> <span m="4527460">hypothesis</span> <span m="4528020">to</span> <span m="4528130">use.</span> </p>
<p><span m="4529140">If</span> <span m="4529310">what</span> <span m="4529460">you've</span> <span m="4529570">got</span> <span m="4529820">doesn't</span> <span m="4530100">work,</span> <span m="4530930">you</span> <span m="4531030">could</span> <span m="4531180">pick</span> <span m="4531330">a</span> <span m="4531360">different</span> <span m="4531800">one,</span> <span m="4533040">but</span> <span m="4533220">better</span> <span m="4533700">to</span> <span m="4533810">pick</span> <span m="4533990">a</span> <span m="4534050">stronger</span> <span m="4534630">one.</span> <span m="4535180">Yeah?</span> </p>
<p><span m="4537031">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="4540930">PROFESSOR: Yes.</span> <span m="4542410">You</span> <span m="4542620">could.</span> <span m="4542890">And</span> <span m="4542990">that</span> <span m="4543140">is</span> <span m="4543220">a</span> <span m="4543290">good</span> <span m="4543520">thing</span> <span m="4543670">to</span> <span m="4543750">do.</span> <span m="4545110">Make</span> <span m="4545770">the</span> <span m="4545930">induction</span> <span m="4546780">hypothesis</span> <span m="4547680">be</span> <span m="4547830">much</span> <span m="4548040">stronger.</span> <span m="4550460">I</span> <span m="4550560">had</span> <span m="4550720">trouble</span> <span m="4551080">proving</span> <span m="4552010">there's</span> <span m="4552180">a</span> <span m="4552240">way</span> <span m="4552400">to</span> <span m="4552500">do</span> <span m="4552670">it</span> <span m="4552810">with</span> <span m="4552960">a</span> <span m="4553050">center</span> <span m="4553430">square</span> <span m="4553700">missing.</span> <span m="4554880">Turns</span> <span m="4555160">out</span> <span m="4555280">to</span> <span m="4555360">be</span> <span m="4555520">easier</span> <span m="4556050">to</span> <span m="4556130">prove</span> <span m="4556450">it</span> <span m="4557710">where</span> <span m="4558100">you</span> <span m="4558280">say</span> <span m="4558530">any</span> <span m="4558850">square</span> <span m="4559190">could</span> <span m="4559300">be</span> <span m="4559390">missing.</span> </p>
<p><span m="4559890">Seems</span> <span m="4560190">like</span> <span m="4560310">this</span> <span m="4560430">should</span> <span m="4560550">be</span> <span m="4560640">harder,</span> <span m="4561060">right?</span> <span m="4561600">We</span> <span m="4561720">had</span> <span m="4561800">a</span> <span m="4561840">hard enough</span> <span m="4562110">time</span> <span m="4562310">just</span> <span m="4562460">showing</span> <span m="4562920">this</span> <span m="4563120">square</span> <span m="4563350">missing</span> <span m="4563650">was</span> <span m="4564060">doable.</span> <span m="4565250">But</span> <span m="4565400">by</span> <span m="4565520">assuming</span> <span m="4565970">something--</span> <span m="4566820">by</span> <span m="4566920">trying</span> <span m="4567120">to</span> <span m="4567180">prove</span> <span m="4567340">a</span> <span m="4567390">harder</span> <span m="4567850">problem,</span> <span m="4568260">assuming</span> <span m="4568570">something</span> <span m="4568830">stronger</span> <span m="4569220">in</span> <span m="4569310">pn,</span> <span m="4570330">it</span> <span m="4570440">gets</span> <span m="4570650">easier</span> <span m="4570960">to</span> <span m="4571040">prove.</span> <span m="4572680">All</span> <span m="4572790">right,</span> <span m="4572960">now</span> <span m="4573050">we</span> <span m="4573130">better</span> <span m="4573350">check</span> <span m="4573560">the</span> <span m="4573640">base</span> <span m="4573930">case,</span> <span m="4575700">p0,</span> <span m="4576140">but</span> <span m="4576250">that's</span> <span m="4576450">easy.</span> <span m="4576900">There's</span> <span m="4576960">only</span> <span m="4577120">one</span> <span m="4577290">square</span> <span m="4577550">Bill</span> <span m="4577750">can be.</span> </p>
<p><span m="4579960">But</span> <span m="4580060">now</span> <span m="4580270">let's</span> <span m="4580520">look</span> <span m="4580670">at</span> <span m="4580760">pn</span> <span m="4581100">implies</span> <span m="4581480">pn</span> <span m="4581770">plus</span> <span m="4582010">1.</span> <span m="4586090">So</span> <span m="4586170">we</span> <span m="4586250">go</span> <span m="4586390">back</span> <span m="4586610">to</span> <span m="4586700">this.</span> <span m="4586930">We've</span> <span m="4587060">got</span> <span m="4587330">a</span> <span m="4587460">2</span> <span m="4587660">to</span> <span m="4587720">the</span> <span m="4587840">n</span> <span m="4588000">plus</span> <span m="4588230">1</span> <span m="4588460">by</span> <span m="4588580">2</span> <span m="4588750">to</span> <span m="4588800">the</span> <span m="4588890">n</span> <span m="4589000">plus</span> <span m="4589260">1</span> <span m="4590530">courtyard.</span> <span m="4592780">And</span> <span m="4593000">now</span> <span m="4593350">Bill</span> <span m="4593800">can</span> <span m="4593950">be</span> <span m="4594520">anywhere.</span> <span m="4598030">Put</span> <span m="4598180">him</span> <span m="4598750">here.</span> <span m="4599750">There's</span> <span m="4599980">Bill.</span> </p>
<p><span m="4602420">Now</span> <span m="4602690">I'm</span> <span m="4602820">going</span> <span m="4602930">to</span> <span m="4602990">place</span> <span m="4603270">my</span> <span m="4603390">first</span> <span m="4603760">l-shaped</span> <span m="4604260">tile</span> <span m="4604550">here,</span> <span m="4605660">in</span> <span m="4605760">the</span> <span m="4605850">other</span> <span m="4606000">three</span> <span m="4606240">regions.</span> <span m="4609000">And</span> <span m="4609170">now</span> <span m="4609360">I</span> <span m="4609450">apply</span> <span m="4609820">pn</span> <span m="4610185">to</span> <span m="4610550">each</span> <span m="4610740">region.</span> <span m="4612710">A</span> <span m="4612840">pn</span> <span m="4613290">says</span> <span m="4613510">I</span> <span m="4613570">can</span> <span m="4613740">do</span> <span m="4614000">it</span> <span m="4614090">with any</span> <span m="4614340">square</span> <span m="4614610">missing.</span> <span m="4614950">So</span> <span m="4615070">I'll</span> <span m="4615170">pick</span> <span m="4615370">this</span> <span m="4615540">one</span> <span m="4615690">out</span> <span m="4615820">here,</span> <span m="4616620">this</span> <span m="4616800">one</span> <span m="4616930">out</span> <span m="4617080">here,</span> <span m="4617540">that</span> <span m="4617760">one</span> <span m="4617920">there,</span> <span m="4618260">that</span> <span m="4618450">one</span> <span m="4618570">there.</span> <span m="4619520">And</span> <span m="4619650">now</span> <span m="4619800">I'm</span> <span m="4619940">done.</span> </p>
<p><span m="4621080">That</span> <span m="4621330">was</span> <span m="4621460">easy.</span> <span m="4622720">No</span> <span m="4622910">extra</span> <span m="4623200">steps.</span> <span m="4625650">Now,</span> <span m="4625780">how</span> <span m="4626090">is</span> <span m="4626200">it</span> <span m="4626310">possible</span> <span m="4626840">that</span> <span m="4626920">it</span> <span m="4626960">was</span> <span m="4627160">easier</span> <span m="4627630">to</span> <span m="4627720">prove</span> <span m="4628050">something</span> <span m="4628340">that</span> <span m="4628430">was</span> <span m="4629340">harder?</span> <span m="4631370">Yeah?</span> </p>
<p><span m="4633706">AUDIENCE: First</span> <span m="4634180">need to be</span> <span m="4634654">in the center</span> <span m="4635128">[INAUDIBLE],</span> <span m="4635602">you're</span> <span m="4636076">putting</span> <span m="4636550">another constraint</span> <span m="4637024">on yourself.</span> </p>
<p><span m="4637500">PROFESSOR: Yes.</span> <span m="4638840">That</span> <span m="4639030">is</span> <span m="4639170">true.</span> <span m="4639470">But</span> <span m="4640280">by</span> <span m="4640770">allowing</span> <span m="4641190">him to</span> <span m="4641240">be</span> <span m="4641440">anywhere,</span> <span m="4642440">I</span> <span m="4642540">could</span> <span m="4642690">have</span> <span m="4642770">started--</span> <span m="4644200">I</span> <span m="4644330">have</span> <span m="4644550">to--</span> <span m="4644700">that's</span> <span m="4644900">a</span> <span m="4644950">possibility.</span> <span m="4648220">See,</span> <span m="4648260">what</span> <span m="4648380">I'm</span> <span m="4648480">trying</span> <span m="4648730">to</span> <span m="4648810">do</span> <span m="4649020">here,</span> <span m="4650420">this</span> <span m="4650650">inductive</span> <span m="4651070">step</span> <span m="4651350">is</span> <span m="4651470">all</span> <span m="4651710">about</span> <span m="4652010">proving</span> <span m="4652370">pn</span> <span m="4653690">implies</span> <span m="4654130">pn</span> <span m="4654480">plus</span> <span m="4654720">1</span> <span m="4654920">is</span> <span m="4655020">true.</span> <span m="4657380">That's</span> <span m="4657630">what</span> <span m="4657700">I'm</span> <span m="4657780">trying</span> <span m="4658000">to</span> <span m="4658060">show.</span> </p>
<p><span m="4660130">Now,</span> <span m="4660470">how</span> <span m="4660810">is</span> <span m="4660920">it</span> <span m="4661020">useful</span> <span m="4661420">for</span> <span m="4661660">me</span> <span m="4661870">if</span> <span m="4662020">pn</span> <span m="4662430">is</span> <span m="4662570">stronger?</span> <span m="4663840">Has</span> <span m="4664070">more?</span> </p>
<p><span m="4665826">AUDIENCE: You</span> <span m="4666290">grow it.</span> <span m="4667127">You prove that</span> <span m="4667504">he can</span> <span m="4667881">be in a</span> <span m="4668260">corner.</span> <span m="4668515">But</span> <span m="4668770">when</span> <span m="4669210">you grow</span> <span m="4669370">it,</span> <span m="4669890">the</span> <span m="4669980">corner</span> <span m="4670425">moves.</span> <span m="4671100">But</span> <span m="4671330">since</span> <span m="4671500">we</span> <span m="4671630">proved</span> <span m="4671850">that it can</span> <span m="4672345">be in any,</span> <span m="4672840">it's fine</span> <span m="4673320">if it</span> <span m="4673800">moves.</span> </p>
<p><span m="4674280">PROFESSOR: Exactly.</span> <span m="4674575">That's exactly</span> <span m="4674870">right.</span> <span m="4676420">ph</span> <span m="4676910">got</span> <span m="4677050">more</span> <span m="4677240">powerful,</span> <span m="4679410">which</span> <span m="4679650">means</span> <span m="4680150">I</span> <span m="4680260">get</span> <span m="4680460">more</span> <span m="4680750">to</span> <span m="4680850">assume</span> <span m="4681220">here.</span> <span m="4682560">In</span> <span m="4682700">the</span> <span m="4682780">recursive</span> <span m="4683270">problem,</span> <span m="4684340">Bill</span> <span m="4684680">can</span> <span m="4684790">be</span> <span m="4684920">anywhere</span> <span m="4685240">now.</span> <span m="4686250">It gives</span> <span m="4686390">me</span> <span m="4686500">more</span> <span m="4686690">power.</span> <span m="4687040">I</span> <span m="4687080">can</span> <span m="4687260">tile</span> <span m="4687630">any</span> <span m="4687890">courtyard</span> <span m="4689480">with</span> <span m="4689640">any</span> <span m="4689850">square</span> <span m="4690100">missing.</span> <span m="4690750">This</span> <span m="4690980">is</span> <span m="4691080">more</span> <span m="4691260">powerful.</span> <span m="4691810">So</span> <span m="4691910">this</span> <span m="4692130">got</span> <span m="4692570">more</span> <span m="4692730">powerful.</span> <span m="4693790">So</span> <span m="4694080">did</span> <span m="4694260">this.</span> </p>
<p><span m="4695440">So</span> <span m="4695600">what</span> <span m="4695760">it</span> <span m="4695900">means</span> <span m="4696210">is</span> <span m="4696300">that my</span> <span m="4696450">tool</span> <span m="4696850">set</span> <span m="4697790">is</span> <span m="4697960">bigger</span> <span m="4698890">with</span> <span m="4699030">a</span> <span m="4699080">stronger</span> <span m="4699460">pn.</span> <span m="4700450">And</span> <span m="4700660">what</span> <span m="4700750">I'm</span> <span m="4700840">trying</span> <span m="4701170">to</span> <span m="4701240">construct</span> <span m="4701510">or</span> <span m="4701780">prove</span> <span m="4702620">got</span> <span m="4702760">harder.</span> <span m="4704240">And</span> <span m="4704440">sometimes,</span> <span m="4706820">if</span> <span m="4706970">I've</span> <span m="4707110">got</span> <span m="4707300">more</span> <span m="4707500">tools,</span> <span m="4708430">it</span> <span m="4708550">becomes</span> <span m="4708840">easier</span> <span m="4709170">to</span> <span m="4709260">prove,</span> <span m="4709590">even</span> <span m="4709780">a</span> <span m="4709830">harder</span> <span m="4710140">thing.</span> </p>
<p><span m="4711750">And</span> <span m="4711880">so</span> <span m="4712010">a</span> <span m="4712050">general</span> <span m="4712430">rule</span> <span m="4712720">with</span> <span m="4712810">induction</span> <span m="4713960">is</span> <span m="4714160">if</span> <span m="4714640">you</span> <span m="4714770">don't</span> <span m="4715300">first</span> <span m="4715590">succeed,</span> <span m="4716120">try,</span> <span m="4716390">try</span> <span m="4716570">again.</span> <span m="4716920">Well,</span> <span m="4717030">the</span> <span m="4717110">rule with</span> <span m="4717290">induction</span> <span m="4717740">is</span> <span m="4717900">if you</span> <span m="4718000">don't</span> <span m="4718190">succeed</span> <span m="4718500">at</span> <span m="4718560">first,</span> <span m="4719130">try</span> <span m="4719310">something</span> <span m="4719600">harder.</span> <span m="4721310">All</span> <span m="4721410">right?</span> <span m="4722070">And</span> <span m="4722350">it's</span> <span m="4722460">amazing,</span> <span m="4723010">but</span> <span m="4723170">it</span> <span m="4723350">actually</span> <span m="4723870">works</span> <span m="4724270">a</span> <span m="4724320">lot</span> <span m="4724610">of</span> <span m="4724670">time,</span> <span m="4725110">as</span> <span m="4725280">it</span> <span m="4725490">did</span> <span m="4725700">here.</span> <span m="4727010">If</span> <span m="4727190">I</span> <span m="4727270">don't</span> <span m="4727470">assume</span> <span m="4727770">something</span> <span m="4728470">strong</span> <span m="4728920">enough,</span> <span m="4729930">and</span> <span m="4730080">pn is</span> <span m="4730460">some</span> <span m="4730890">little</span> <span m="4731130">weak</span> <span m="4731410">thing</span> <span m="4731650">like</span> <span m="4731850">Bill</span> <span m="4732090">just</span> <span m="4732320">in</span> <span m="4732930">the</span> <span m="4733010">center,</span> <span m="4734100">it's</span> <span m="4734300">not</span> <span m="4734510">enough</span> <span m="4734750">to</span> <span m="4734860">go</span> <span m="4735040">anywhere.</span> </p>
<p><span m="4737120">But</span> <span m="4737230">if</span> <span m="4737320">I</span> <span m="4737390">could</span> <span m="4737520">assume</span> <span m="4737830">a</span> <span m="4737880">lot</span> <span m="4738120">more,</span> <span m="4738520">like</span> <span m="4738750">Bill</span> <span m="4738940">could</span> <span m="4739050">be</span> <span m="4739270">anywhere</span> <span m="4739860">he</span> <span m="4740030">pleases.</span> <span m="4741680">Then</span> <span m="4741930">I</span> <span m="4741990">could</span> <span m="4742150">prove</span> <span m="4742380">lots</span> <span m="4742650">of</span> <span m="4742770">things</span> <span m="4743520">here.</span> <span m="4746200">So</span> <span m="4746350">it's all</span> <span m="4746690">the</span> <span m="4746820">art--</span> <span m="4747100">and</span> <span m="4747190">you're</span> <span m="4747350">going to learn</span> <span m="4747660">this over</span> <span m="4747790">the</span> <span m="4747890">next</span> <span m="4748090">several</span> <span m="4748320">weeks--</span> <span m="4748530">it's</span> <span m="4748670">all</span> <span m="4748980">the</span> <span m="4749120">art</span> <span m="4750050">of</span> <span m="4750190">what's</span> <span m="4750380">your</span> <span m="4750540">induction</span> <span m="4750910">hypothesis.</span> <span m="4752650">Picking</span> <span m="4752850">a</span> <span m="4752970">good</span> <span m="4753180">one,</span> <span m="4753400">life</span> <span m="4753650">is</span> <span m="4753790">easy.</span> <span m="4754110">Picking</span> <span m="4754970">the</span> <span m="4755110">wrong</span> <span m="4755400">one,</span> <span m="4756010">very</span> <span m="4756230">painful,</span> <span m="4757140">as</span> <span m="4757300">you'll</span> <span m="4757420">see</span> <span m="4757580">with</span> <span m="4757710">beaver</span> <span m="4757980">flu.</span> <span m="4759830">OK,</span> <span m="4760170">that's</span> <span m="4760400">it</span> <span m="4760510">for</span> <span m="4760630">today.</span> </p>
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<h3 class="subsubhead">Video</h3>
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<h3 class="subsubhead">Caption</h3>
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<li>English-US (<a href="../../../contents/video-lectures/lecture-2-induction/z8HKWUWS-lA.srt">SRT</a>)</li>
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